Linear Functions: Slope and Intercept

A linear relationship changes at a steady rate. Whenever the input increases by the same amount, the output changes by the same amount. Its points lie on a straight line.

Check equal input steps

Suppose a table has inputs 0, 1, 2 and 3, with outputs 5, 8, 11 and 14. The input increases by 1 each time. The output increases by 3 each time, so the rate is 3 per input unit.

A steady decrease works too. A 20 cm candle loses 2 cm of height each hour. Its heights after 0, 1, 2 and 3 hours are 20, 18, 16 and 14 cm. Each change is negative 2, so the rate is $-2$ cm per hour.

Check all the intervals. Outputs 0, 1, 4 and 9 for inputs 0, 1, 2 and 3 change by 1, then 3, then 5. That changing rate makes the relationship nonlinear. Doubling each time, as in 1, 2, 4 and 8, is also nonlinear: multiplying by a fixed number is different from adding a fixed amount.

Check unequal input steps

Different gaps between inputs need a fair comparison. Divide each output change by its matching input change:

For inputs 1, 2, 4 and 7, suppose the outputs are 3, 5, 9 and 15.

The output changes are unequal because the input gaps are unequal. All three rates are 2, so these points lie on one straight line. Comparing just the output changes would give the wrong decision.

Find the steady change in words and patterns

A gym charges $25 to join and $10 per month. The totals after 1, 2 and 3 months are $35, $45 and $55. The joining fee is paid once; each extra month adds $10.

A row of matchstick squares has a similar pattern. One square uses 4 matches. Each new square shares one side, so it needs 3 more matches. Two squares use 7, and three use 10. For $n$ squares, the rule is $M=3n+1$.

Only whole numbers of squares belong to this pattern. Plot its separate points without joining them: a point for 1.5 squares would describe something the pattern does not include. For quantities such as time and candle height, values between whole numbers make sense, so a line can join the points.

Linear and proportional

A proportional relationship is a linear relationship that passes through the origin $(0,0)$. It has the form $y=kx$, with no added starting amount.

The gym costs are linear but not proportional. Two months cost $45, which is not twice the $35 cost of one month. The fixed joining fee stops the total from doubling when the number of months doubles.

The slope of a line measures its direction and steepness. It tells you how much the height changes for a horizontal move. Slope is also called gradient.

Rise divided by run

Choose two points whose coordinates you can read exactly. Move from the left point toward the right point. The run is the horizontal change. The rise is the vertical change, including its sign.

For points $(1,2)$ and $(4,8)$, the run is $4-1=3$ and the rise is $8-2=6$. The slope is $6\mathrm{/}3=2$: the line rises 2 units for every unit to the right.

Use grid corners when the line passes through them. Some lines have no grid-corner points on a unit grid. On $y=x+1.5$, clearly marked points such as $(0,1.5)$ and $(2,3.5)$ still give exact coordinates.

Read the direction and steepness

For a line through $(0,5)$ and $(2,1)$, the rise is $-4$ and the run is 2. Its slope is $-2$. The minus sign records the downward direction.

To compare steepness on matching axes, compare the sizes of the slopes without their signs. The absolute value $\mathrm{\mid }m\mathrm{\mid }$ gives this size. Slope $-3$ is steeper than slope 2 because 3 is greater than 2. The negative sign does not make the line less steep.

A horizontal line has zero rise, so its slope is zero. A vertical line has zero run. Its slope is undefined because division by zero has no value.

Use a larger slope triangle

A slope triangle shows the horizontal and vertical moves between two points. On a straight line, triangles of different sizes have the same angles. They are similar, so their matching side lengths have the same ratio.

A rise of 2 over a run of 3 gives $2\mathrm{/}3$. Doubling the triangle gives a rise of 4 over a run of 6, still $4\mathrm{/}6=2\mathrm{/}3$. This is why choosing points farther apart keeps the same slope. A fractional slope such as $3\mathrm{/}4$ is useful for plotting: move right 4 and up 3.

Read the scale before counting

Count axis units, not just squares. Suppose one square is 1 unit across but 10 units up. A rise of 3 squares and run of 1 square gives slope $30\mathrm{/}1=30$. Changing an axis scale also changes how steep a line looks on the page.

For a ramp, use its horizontal distance as the run. A ramp rising 0.5 m over a horizontal distance of 6 m has slope $0.5\mathrm{/}6=1\mathrm{/}12\approx 0.083$. The length along the ramp is a sloping side, so it is not the run. Measure rise and run in the same units when comparing real slopes.

Coordinates give the rise and run without counting squares. An ordered pair lists the horizontal coordinate first and the vertical coordinate second. Keep each point’s coordinates together when you subtract.

Use the slope formula

For two points $(x_1,y_1)$ and $(x_2,y_2)$ with different horizontal coordinates,

The top is the change in height; the bottom is the horizontal change. For $(-1,6)$ and $(3,-2)$:

1. Find the vertical change: $-2-6=-8$.
2. Find the horizontal change: $3-(-1)=4$.
3. Divide: $m=-8\mathrm{/}4=-2$.

The line falls from left to right, which agrees with the negative result. Write negative coordinates in parentheses before simplifying. Subtracting negative 1 adds 1; it does not subtract another 1.

Keep the same order in both differences

Either point may come first. Reversing both differences gives

Reversing just one difference changes the sign incorrectly. Also keep the vertical change on top: run divided by rise gives the reciprocal when both changes are nonzero.

A slope need not be a whole number. For $(-2,5)$ and $(4,1)$:

Simplify the fraction. It says that for a move of 3 units to the right, the line drops 2 units.

Notice equal coordinates

Points $(-3,4)$ and $(5,4)$ have equal heights. Their vertical change is zero, so $m=0\mathrm{/}8=0$. They lie on a horizontal line.

Points $(2,-1)$ and $(2,7)$ have the same horizontal coordinate. Their run is zero, so the slope is undefined. They lie on a vertical line. Zero slope and undefined slope describe different directions.

Work from a table and attach units

A repair bill is $70 for 2 hours, $115 for 5 hours and $175 for 9 hours. Compare neighboring rows:

Both rates are $15 per hour. The units come from output units divided by input units. A dollars-versus-hours table gives dollars per hour, not hours per dollar.

Equal slopes between neighboring points, ordered by their horizontal coordinates, show that they lie on one nonvertical line. Check every interval. Points all sharing one horizontal coordinate instead lie on a vertical line.

Know what two readings tell you

A car is 120 km from home after 2 hours and 300 km from home after 5 hours. Its distance from home increased at an average rate of $180\mathrm{/}3=60$ kilometers per hour over that interval. The readings do not show each moment of the trip. This average is its speed throughout only when it travels steadily straight away from home.

A straight-line equation connects each input to its output. In slope-intercept form, its two numbers tell you the rate of change and where the line crosses the vertical axis.

Here $m$ is the slope. The y-intercept is $b$: putting $x=0$ gives $y=b$, so the crossing point is $(0,b)$. Many countries write the same form as $y=mx+c$, using $c$ for the intercept.

Keep the signs with the numbers

In $y=3x-4$, the slope is 3 and the intercept is $-4$. The subtraction means that negative 4 is added to $3x$.

The coefficient multiplying the input is the slope, wherever that term appears. A missing coefficient before $x$ means 1; a minus sign alone means $-1$. A missing constant term means an intercept of zero.

Solve for the output first

In standard form, an equation is written $Ax+By=C$. The letters $A$, $B$ and $C$ are fixed numbers. When $B\mathrm{\ne }0$, rearrange to make $y$ stand alone before reading the slope.

For $2x+3y=12$:

1. Subtract $2x$ from both sides: $3y=-2x+12$.
2. Divide every term by 3: $y=-\frac{2}{3}x+4$.

The slope is $-2\mathrm{/}3$ and the intercept is 4. Reading 2 as the slope from the original equation misses both the rearrangement and the division.

For $4y-8x=20$, add $8x$ and then divide by 4:

The constant must be divided too. Keeping 20 as the intercept would change the line.

Recognize the form of the relationship

Every nonvertical straight line can be written as $y=mx+b$. A vertical line such as $x=3$ has undefined slope and cannot be written in that form.

After simplification, a linear equation has its variables only to the first power. There is no product of the variables and no input variable in a denominator. The rule $y=(x+6)\mathrm{/}2$ is linear because it becomes $y=x\mathrm{/}2+3$. The rules $y=x^2+1$ and $y=3\mathrm{/}x$ are nonlinear.

Read function notation and test points

Function notation $f(x)$ names the output for input $x$. It does not mean that $f$ is multiplied by $x$. If $f(x)=4x-1$, then $f(2)=4(2)-1=7$. The point $(2,7)$ is on the graph.

To test any point, substitute its horizontal coordinate for the input and compare the result with its vertical coordinate. For $(4,5)$ on $y=2x-3$, the calculation $2(4)-3=5$ matches. For $(1,2)$, it gives $-1$, so that point is off the line.

An equation gives points you can plot. Two distinct points fix a straight line. A third point helps you catch a calculation, plotting or scale error before drawing it.

Start with the intercept and slope

To graph $y=\frac{2}{3}x-1$:

1. Plot the y-intercept $(0,-1)$.
2. Use the slope: move right 3 and up 2 to $(3,1)$.
3. Repeat the move to $(6,3)$.
4. Draw a ruled line through the points and label it with its equation.

For a negative slope, move down as you go right. On $y=-2x+4$, start at $(0,4)$, then move right 1 and down 2. This gives $(1,2)$ and $(2,0)$. Moving left 1 and up 2 also stays on the line, because both changes reverse sign.

Make a table of values

Choose convenient inputs and calculate each output. For $y=3x-2$:

Plot $(-1,-5)$, $(0,-2)$ and $(2,4)$. If your points do not line up, inspect the calculations, coordinate order and axis scales.

Use the axis crossings

At the x-intercept, the output is zero. For $y=2x-6$, set $y=0$ and solve:

The crossing point is $(3,0)$. The input 3 is also called the function’s zero or root. A line with nonzero slope has one x-intercept. A horizontal line above or below the x-axis has none; the line $y=0$ lies along that axis.

The intercept method is handy for $3x+4y=12$. Set $x=0$ to get $(0,3)$. Set $y=0$ to get $(4,0)$. Join the two points. If both intercepts are the origin, they give only one point, so calculate another point.

An equation such as $x+y=5$ describes points whose coordinates add to 5. The points $(0,5)$ and $(5,0)$ give its graph, which is also $y=-x+5$.

Draw horizontal and vertical lines

The line $y=3$ stays at height 3 for every input. It is horizontal, with slope zero. The line $x=3$ keeps its horizontal coordinate fixed. It is vertical, with undefined slope. The x-axis has equation $y=0$; the y-axis has equation $x=0$.

A vertical line gives many outputs for the same input, so it is not a function graph. Nonvertical lines are function graphs: positive slopes give increasing functions, negative slopes give decreasing functions, and zero slope gives a constant function.

Label axes and read their units carefully. Choose ranges that show the needed points. An intercept of 1.5 belongs halfway between 1 and 2; it need not land on a gridline. Check your finished line with a point you did not use to draw it.

Two lines can start at different heights and still change at the same rate. Comparing their slopes and intercepts helps you separate steepness from position.

Change the intercept

Keep the slope fixed at 2 and compare $y=2x$, $y=2x+3$ and $y=2x-1$. Adding 3 raises every output by 3. Subtracting 1 lowers every output by 1. These moves slide the line up or down without changing its slope.

A greater intercept does not make a line steeper. The lines $y=2x+5$ and $y=2x$ both rise 2 units for each unit to the right. Their outputs stay 5 units apart.

Change the slope

Keep the intercept fixed at 1. Each line in $y=mx+1$ passes through $(0,1)$. Changing the slope turns the line about that point. A greater absolute value of the slope gives a steeper line on matching axes.

The lines $y=2x+1$ and $y=-2x+1$ have equal steepness but opposite directions. They are mirror images in the y-axis. The intercept stays fixed while the sign of the slope changes.

Distinguish parallel lines from the same line

Two different nonvertical lines are parallel exactly when they have equal slopes. Their intercepts must differ. If both the slopes and intercepts match, the equations describe the same line.

Rearrange equations before comparing them. For $6x-2y=8$:

It is parallel to $y=3x-1$ because both slopes are 3 and the intercepts differ. But $4x-2y=-2$ rearranges to $y=2x+1$, so those two equations name one line.

Different vertical lines, such as $x=1$ and $x=4$, are also parallel. Compare their fixed horizontal coordinates; they have no numerical slopes to compare.

Match an equation to a graph

Use the sign of the slope, its size, and the y-intercept together. Among $y=-x+3$, $y=2x+3$ and $y=2x-1$, the first is the falling line. The first two share the crossing $(0,3)$. The last two are parallel.

When the intercept is zero, the line passes through the origin and is proportional. The lines $y=0.5x$, $y=x$ and $y=3x$ all start there. On $y=x$, the two coordinates of each point are equal.

Compare a table with an equation

Function A is $y=4x+10$. Function B has inputs 0, 2 and 4, with outputs 3, 13 and 23. B’s slope is $10\mathrm{/}2=5$ and its intercept is 3.

A starts higher, but B grows faster. A greater starting value does not guarantee a greater output at every input. To compare their outputs at a particular input, calculate both values.

A straight-line graph contains the information needed for its equation. The y-axis crossing gives the intercept. Two readable points give the slope. Read the axis labels before using either feature.

Read the intercept and calculate the slope

Suppose a line crosses the y-axis at $(0,-3)$ and passes through $(2,1)$. The intercept is $b=-3$. Use the two points to find the slope:

Put the two values into slope-intercept form:

Check a different point from the graph. If $(3,3)$ is marked, substitution gives $2(3)-3=3$, as needed. This catches a line drawn or read incorrectly as well as arithmetic slips.

For a falling line through $(0,4)$ and $(6,0)$:

Its equation is $y=-\frac{2}{3}x+4$. A falling line must have a negative slope. Using $2\mathrm{/}3$ would make its outputs increase as the input increases.

Work out an intercept that is hard to read

A line may cross between gridlines, or its y-axis crossing may be outside the visible window. Choose two points you can read exactly instead of guessing the crossing.

For a line through $(1,2)$ and $(3,5)$:

1. Calculate the slope: $m=(5-2)\mathrm{/}(3-1)=3\mathrm{/}2$.
2. Start the equation: $y=\frac{3}{2}x+b$.
3. Substitute $(1,2)$: $2=\frac{3}{2}(1)+b$.
4. Subtract $3\mathrm{/}2$: $b=1\mathrm{/}2$.

The equation is $y=1.5x+0.5$. The second point checks it: $1.5(3)+0.5=5$. The height 2 at input 1 was not the intercept; the intercept is the height at input zero.

Exact points need not be grid corners. A labeled point with a fractional coordinate can be exact. If a graph only lets you estimate the coordinates, the equation you find will also be an estimate.

Use the labeled scale

Suppose the y-axis increases by 2 units per square. A crossing three squares above the origin gives an intercept of 6. Writing 3 would use the number of squares instead of the coordinate value.

Read rise and run in their own axis units too. One axis may count in ones and the other in fives. Choose points far enough apart to read clearly, and write their coordinates before calculating the slope.

Handle horizontal and vertical lines

A horizontal line through $(0,-2)$ has equation $y=-2$. Its height stays fixed. A vertical line through $(5,0)$ has equation $x=5$ because every point has horizontal coordinate 5.

For a nonvertical line, keep the roles of the two numbers clear: slope multiplies the input and the y-intercept is added. An x-axis crossing is not the y-intercept.

A slope gives a line’s direction, and a point fixes its position. With two distinct points, first calculate the direction from their coordinate changes. Then find the intercept.

Given a slope and a point

For slope 3 through $(2,11)$, start with $y=3x+b$. The point tells you that the output is 11 when the input is 2:

The equation is $y=3x+5$. The point’s height, 11, is not the intercept because its input is not zero.

Point-slope form is another way to use the same information:

For slope $-2$ through $(4,1)$, it gives $y-1=-2(x-4)$. Expand the parentheses to get $y-1=-2x+8$, then add 1: $y=-2x+9$.

With a negative coordinate, subtract the whole coordinate. For slope $1\mathrm{/}3$ through $(-3,5)$:

Expanding gives $y=\frac{1}{3}x+6$. The plus sign inside the parentheses comes from subtracting negative 3.

Given two points

Find the equation through $(-2,-1)$ and $(4,8)$.

1. Find the slope: $m=\frac{8-(-1)}{4-(-2)}=\frac{9}{6}=\frac{3}{2}$.
2. Substitute the first point: $-1=\frac{3}{2}(-2)+b=-3+b$.
3. Solve for the intercept: $b=2$.
4. Write the equation: $y=\frac{3}{2}x+2$.

Check with the other point: $\frac{3}{2}(4)+2=8$. Either point gives the same intercept. Using the second point for the check tests information beyond the substitution you already used.

Given a table

Suppose inputs 3, 6 and 9 have outputs 10, 16 and 22. The rate is $6\mathrm{/}3=2$. Using the first row gives $10=2(3)+b$, so $b=4$. The equation is $y=2x+4$, and the final row checks: $2(9)+4=22$.

The first output in a table is the intercept only when its input is zero. If more rows are available, check that they follow the same rate before choosing one linear rule.

Given a parallel line

A line parallel to $y=4x-7$ has slope 4. To make it pass through $(1,6)$, solve $6=4(1)+b$. This gives $b=2$ and $y=4x+2$. The original line supplies the slope; the new point supplies the new position.

Special cases and equivalent forms

Points with the same height give a horizontal line. Through $(-1,4)$ and $(6,4)$, the equation is $y=4$. Points with the same horizontal coordinate give a vertical line. Through $(3,-2)$ and $(3,5)$, it is $x=3$. A repeated point alone does not fix a direction.

The same line can have different equation forms. Multiply $y=\frac{3}{2}x+2$ by 2 to get $2y=3x+4$. Rearranging gives $3x-2y=-4$. All three forms describe the same points because each rearrangement keeps both sides equal.

A model uses an equation to describe a situation. A linear model fits a constant rate of change. To build one, name the input and output, give their units, and connect the starting amount to the amount added or lost per unit.

Turn a description into an equation

Renting a car for one day costs $40 plus $0.20 per mile. Let $d$ be miles driven and $C$ be the total cost in dollars. The fixed charge is 40, and each mile adds 0.20:

For 150 miles, $C=0.20(150)+40=70$. The total is $70. Both amounts in the equation use dollars. A rate of 20 would mean $20 per mile, not 20 cents per mile.

A useful order is to identify the quantities first, then the rate, then the output at zero input. Write the rule as output equals rate times input plus starting value. Test it with a value you know.

Model a decrease

A tank starts with 600 liters and drains 25 liters per minute. Let $t$ be minutes and $V$ be liters remaining:

The rate is negative because the volume decreases. To find when the tank empties, set $V=0$:

The model fits times from 0 through 24 minutes. Extending the line beyond 24 would predict negative water, so the draining rule stops at the empty tank.

Find the rate and starting value from data

A plumber charges $155 for 2 hours and $275 for 5 hours. The extra 3 hours add $120, so the hourly rate is

Let $h$ be hours and $C$ be dollars. Start with $C=40h+b$. Substitute the 2-hour bill:

The model is $C=40h+75$. It also gives the 5-hour bill: $40(5)+75=275$. Swapping the fee and rate would give $C=75h+40$, which fails the known bills.

Read a model from a graph or pattern

A fare graph passes through $(0,3.50)$ and $(10,23.50)$, with kilometers on the horizontal axis and dollars on the vertical axis. The slope is $20\mathrm{/}10=2$ dollars per kilometer. The intercept is 3.50, so the fare model is $F=2k+3.50$ for $k$ kilometers.

A seating pattern can also give a model. One rectangular table seats 6 people. Each extra table in a joined row adds 4 seats, so the rule is $S=4n+2$ for $n$ tables. It gives 2 at input zero, even though a row of zero tables is not an actual seating arrangement. The intercept locates the line without describing a real pattern stage.

Check whether one rate fits

One straight-line rule needs one constant rate over the inputs being modeled. A discount after a large order changes the price per item. A phone plan with an included allowance changes its charge after that allowance. Those whole situations need more than one constant-rate rule.

A number in a model has meaning because of the quantities and units around it. Explain a slope as a change per input unit. Explain an intercept as the output when the input is zero.

Describe a rate in words

In $C=40h+75$, $C$ is a bill in dollars and $h$ is work time in hours. The slope 40 means that each extra hour adds $40 to the bill. It is a rate in dollars per hour.

For $V=600-25t$, with liters of water remaining after $t$ minutes, the slope is $-25$ liters per minute. A clear interpretation is: “Each additional minute reduces the water volume by 25 liters.” The minus sign supplies the direction; the units say which change is being measured.

Saying “the slope is 40” identifies a number but leaves out its meaning. Include the output, the input unit, and whether the output increases or decreases.

Explain the output at zero

In the bill model, zero hours gives a cost of $75. That is the fixed charge before hourly work is added. In the tank model, zero minutes gives 600 liters, the starting volume.

A table may start after time zero. Its first output then belongs to its first recorded time. Find the intercept from the rate and a known point; do not simply call the first reading the starting value.

Separate a temperature change from a reading

The conversion rule is $F=1.8C+32$, where $C$ is a Celsius reading and $F$ is the Fahrenheit reading.

The slope 1.8 means a rise of 1 °C gives a rise of 1.8 °F. The intercept 32 means that a reading of 0 °C equals 32 °F. A temperature change and a temperature reading are different uses of the rule. For example, a reading of 100 °C becomes $1.8(100)+32=212$ °F; the 32 is an offset between the scales.

Ask whether zero input is a real case

Suppose $P=0.4d+3$ models pizza prices in dollars for diameters $d$ between 20 and 40 cm. The intercept is 3. Algebraically, it is the predicted price at diameter zero. A zero-diameter pizza is not an item being sold, so this intercept is not an actual price in that range.

The intercept still fixes the line’s position. Its mathematical meaning is always the output at zero input; a real starting-value meaning depends on whether zero belongs to the situation.

Read motion and compare units

On a distance-traveled graph, slope gives speed. A straight segment from $(0,0)$ to $(4,1000)$, in minutes and meters, gives $1000\mathrm{/}4=250$ meters per minute. Two readings alone give the average speed between them; a straight segment describes a constant speed over that interval. Distance from home can change differently if the traveler turns back.

Use matching units before comparing rates. A rate of 60 kilometers per hour equals 1 kilometer per minute. The number 60 is larger, but these are the same speed.

For a linear model with zero intercept, doubling the input doubles the output. A nonzero fixed fee breaks that proportional scaling. Under $C=40h+75$, 2 hours cost $155 and 4 hours cost $235, not $310.

A linear model lets you calculate an output, work backward to an input, or compare two choices. Keep the meaning of the input and the model’s usable range in view while you calculate.

Predict an output or find an input

For $C=40h+75$, a 3.5-hour job costs $40(3.5)+75=215$ dollars. Multiply by the hours first, then add the fixed charge once.

To find how much time a $300 budget covers, set the output to 300:

The budget covers up to 5.625 hours. The decimal part is a fraction of an hour: $0.625\times 60=37.5$ minutes. This is 5 hours and 37.5 minutes, not 5 hours and 62.5 minutes. Checking $40(5.625)+75=300$ confirms the cost.

A graph gives the same connection. Start at the known value on one axis, move straight to the line, and then move to the other axis to read the result. A reading between gridlines is an estimate. Use the equation for an exact calculation.

Compare two plans

Phone plan A costs $30 per month plus $0.05 per minute. Plan B costs $10 per month plus $0.15 per minute. For $x$ minutes:

Find where the costs agree:

Each plan then costs $40. Their graphs cross at $(200,40)$. Before that input, B is cheaper; after it, A is cheaper. At 100 minutes, the costs are $35 and $25. At 300 minutes, they are $45 and $55.

The lower fixed charge helps B at low usage. The lower rate helps A as usage grows. On a common graph, the cheaper plan has the lower line at the input being compared.

Decide how far to trust a prediction

Interpolation predicts within the range of recorded data. Extrapolation predicts outside that range. A seedling measured from week 0 through week 6 follows $h=2w+5$, with height in centimeters. Predicting 13 cm at week 4 stays within the measured period. Predicting 205 cm at week 100 extends the same growth rate far beyond it. The calculation is correct, but the plant may not keep growing at that rate.

Interpolation can still be imperfect when real measurements vary. Extrapolation needs particular care because the pattern has to continue beyond the data used to build it.

Respect limits and whole counts

The tank model $V=600-25t$ reaches zero at 24 minutes. At 30 minutes it gives negative 150 liters, so it no longer describes the tank. For tickets, people or pattern stages, inputs must be whole counts.

Check the units, sign and size of your answer, then substitute it back. If a negative time solves the equation, it may refer to a time before the model begins. Decide whether that input fits the situation before reporting it as a usable answer.