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Rates of change and intercepts, read from an equation, a table or a graph.
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In this lesson you compare two functions given in different forms, one as an equation and another as a table, a graph or a description, by finding the rate of change and the initial value of each. The form is a costume: the same function can wear any of them, and pulling the same two numbers out of every form is what makes two functions comparable.
You know that a function gives exactly one output $y$ for each input $x$. You can find the slope of a line from two points: change in $y$ divided by change in $x$. You know the form $y = mx + b$, where $m$ is the slope and $b$ is the value of $y$ when $x = 0$. You can read a table and plot points on a graph. This lesson uses all of that to compare two functions that are described in different ways, such as one by an equation and the other by a table.
| Term | What it means |
|---|---|
| Function | A rule that gives exactly one output for each input. |
| Representation | A way of describing a function: an equation, a table, a graph or words. |
| Rate of change | How much the output changes for each increase of $1$ in the input: change in $y$ divided by change in $x$. For a line it is the slope. |
| Initial value ($y$-intercept) | The output when the input is $0$; where the graph crosses the $y$-axis. |
| Increasing / decreasing | A function is increasing when $y$ goes up as $x$ goes up (positive rate), and decreasing when $y$ goes down (negative rate). |
| Steeper | Changing faster. Of two lines, the steeper one has the rate farther from $0$, whether positive or negative. |
A linear function can be written as an equation, listed in a table, drawn as a graph or told in words. These are four costumes for the same thing. Whatever the costume, a linear function is decided by just two numbers:
To compare two functions, pull those two numbers out of each one and then compare numbers with numbers. Here is where to find them:
| Form | Rate of change | Initial value |
|---|---|---|
| Equation $y = mx + b$ | the number $m$ multiplying $x$ | the number $b$ |
| Table | change in $y$ ÷ change in $x$ between two rows | $y$ in the row where $x = 0$ |
| Graph | rise ÷ run between two points on the line | where the line crosses the $y$-axis |
| Words | the amount "per" or "each" | the amount "to start" or "up front" |
For example, Function A is $y = 3x + 7$ and Function B has the table $x$: $0, 1, 2$ and $y$: $2, 7, 12$. A's rate is $3$ and its initial value is $7$. B's rate is $7 - 2 = 5$ and its initial value is $2$. So B grows faster, while A starts higher. Neither is simply "bigger": the two numbers answer two different questions.
Another way: picture
Draw both functions on one grid. The initial value is where each line starts on the $y$-axis. The rate is how steeply each line climbs. A line that starts lower but climbs more steeply will cross the other one and then stay above it.
Another way: story
Two friends save money. Ana has $\$7$ and adds $\$3$ a week. Ben has $\$2$ and adds $\$5$ a week. Ana is ahead at the start; Ben saves faster. After $2$ weeks Ana has $\$13$ and Ben $\$12$; after $3$ weeks Ana has $\$16$ and Ben $\$17$, and from then on Ben stays ahead.
A table does not have to go up by $1$ at a time, and it does not have to start at $x = 0$. Look at this one:
| $x$ | $2$ | $5$ | $8$ |
|---|---|---|---|
| $y$ | $9$ | $21$ | $33$ |
From $x = 2$ to $x = 5$, $x$ goes up by $3$ and $y$ goes up by $12$. So the rate is $12 \div 3 = 4$, not $12$. Always divide by the change in $x$.
Check that it really is linear: from $x = 5$ to $x = 8$ the change is again $12$ over $3$, so the rate is the same. A linear function has the same rate between any two rows.
To find the initial value, walk back to $x = 0$. The first row is at $x = 2$, two units away, and each unit back takes $4$ off $y$. So at $x = 0$, $y = 9 - 2 \times 4 = 1$. The equation is $y = 4x + 1$, and you can test it on any row: $4 \times 8 + 1 = 33$.
On a graph, pick two points where the line crosses grid corners exactly, so you can read their coordinates without guessing. Go from the left point to the right point: count the run (across) and the rise (up is positive, down is negative). The rate is rise divided by run.
A line through $(0, 10)$ and $(4, 2)$ falls $8$ while it runs $4$, so its rate is $-8 \div 4 = -2$. Its initial value is $10$, where it meets the $y$-axis.
When you compare a falling line with another function, be careful with the word "greater". The rate $-2$ is greater than $-3$, because it is to the right on the number line. But a line with rate $-3$ is steeper: it falls $3$ for every $1$ across instead of $2$. So:
Read the question to see which one it wants. In a story about a draining tank, "drains faster" means steeper, so you compare $3$ and $2$, not $-3$ and $-2$.
Sometimes the question is not which function grows faster but which is larger at a particular input. Then put that input into both. With Function A, $y = 3x + 7$, and Function B, $y = 5x + 2$:
The function with the larger initial value wins at the start, and the function with the larger rate wins in the long run. Somewhere in between they are equal: here at $x = 2.5$, where both give $14.5$. You will learn to find that crossing point exactly when you study systems of equations. For now, a table of both functions side by side shows where the lead changes hands, and it is often the quickest way to see what is happening.
Why each move is allowed. A linear function changes by the same amount for each step in $x$, so any two points give the same rate. That is why you can choose whichever two rows or points are easiest. And any linear function can be written as $y = mx + b$, so once you have $m$ and $b$ you have the whole function, whatever costume it came in.
How to check. Test your equation on a row of the table or a point on the graph that you did not use. Sketch both lines: the one with the larger rate should look steeper, and the one with the larger initial value should start higher on the $y$-axis. If the story says something drains, falls or is used up, its rate must be negative.
The federal minimum wage in the United States is $\$7.25$ an hour. Suppose Job A pays exactly that, so pay is $P = 7.25h$ dollars for $h$ hours: rate $7.25$, initial value $0$. Job B gives a $\$50$ signing bonus, and its pay stub shows $\$50$ after $0$ hours, $\$110$ after $10$ hours and $\$170$ after $20$ hours.
Job B's rate is $(110 - 50) \div 10 = 6$ dollars an hour, with initial value $50$. So Job B starts ahead by $\$50$ but earns $\$1.25$ less per hour. The gap closes by $\$1.25$ each hour, so it takes $50 \div 1.25 = 40$ hours for Job A to catch up. A student who will work only $30$ hours all summer should take Job B; one who will work $100$ hours should take Job A.
Weather reports in the US use degrees Fahrenheit; most of the world uses degrees Celsius. The rule is $F = 1.8C + 32$: rate $1.8$, initial value $32$, because water freezes at $0\,^{\circ}$C, which is $32\,^{\circ}$F. Scientists also use kelvins, and a table of Celsius against kelvins reads $0 \to 273.15$, $10 \to 283.15$, $20 \to 293.15$: rate $10 \div 10 = 1$, initial value $273.15$.
Comparing rates tells you that a warming of $1$ degree Celsius is a warming of $1.8$ degrees Fahrenheit but only $1$ kelvin. So when a forecast says it will be $10\,^{\circ}$C warmer tomorrow, that is $18\,^{\circ}$F warmer: the Fahrenheit scale changes almost twice as fast.
Plan A is described in words: $\$15$ a month plus $\$10$ for each gigabyte of data. Plan B is shown on a graph whose line passes through $(0, 35)$ and $(4, 55)$. Plan A is $C = 10g + 15$. Plan B's rate is $(55 - 35) \div 4 = 5$ dollars per gigabyte, so it is $C = 5g + 35$.
For a light user with $2$ GB a month, A costs $\$35$ and B costs $\$45$. For a heavy user with $8$ GB, A costs $\$95$ and B costs $\$75$. The cheaper plan depends on how much data you use, and the two numbers, rate and starting fee, are exactly what you need to decide.
Forgetting to divide by the change in $x$. If a table goes $x = 0, 2, 4$ and $y$ goes up by $10$ each row, the rate is $5$, not $10$.
Reading the first row as the intercept. The initial value is $y$ when $x = 0$. If the table starts at $x = 3$, the first $y$ is not the intercept.
Comparing starting values to decide which grows faster. A bigger intercept only means a head start. Growth is the rate.
Losing the sign on a falling line. A line that goes down from left to right has a negative rate.
Mixing up "greater" and "steeper". $-4$ is less than $-1$, but a line with rate $-4$ is steeper.
Function A is $y = 4x + 1$. Function B has $x$: $0, 1, 2, 3$ and $y$: $6, 8, 10, 12$. Read A's rate.
$m_A = 4$
In $y = mx + b$ the rate is the number multiplying $x$.
Read A's initial value.
$b_A = 1$
When $x = 0$, $y = 4 \times 0 + 1 = 1$.
Find B's rate from two rows.
$m_B = \dfrac{8 - 6}{1 - 0} = 2$
Each step of $1$ in $x$ adds $2$ to $y$.
Read B's initial value.
$b_B = 6$
The table has a row with $x = 0$, and there $y = 6$.
Compare the rates and the initial values.
$m_A = 4 > m_B = 2, \qquad b_B = 6 > b_A = 1$
A grows twice as fast, but B starts $5$ higher.
Function A's graph passes through $(0, 10)$ and $(4, 2)$. Function B is $y = -3x + 12$. Find A's change in $y$.
$2 - 10 = -8$
Going from the first point to the second, the line falls $8$.
Find A's change in $x$.
$4 - 0 = 4$
Subtract the $x$-coordinates in the same order.
Divide to get A's rate.
$m_A = \dfrac{-8}{4} = -2$
A falls $2$ for each $1$ across.
Read B's rate.
$m_B = -3$
The number multiplying $x$ in B's equation.
Decide which falls faster.
$|-3| = 3 > |-2| = 2$
Steepness is the size of the rate, ignoring the sign, so B is steeper.
Compare the initial values.
$b_A = 10, \qquad b_B = 12$
A crosses the $y$-axis at $10$ and B at $12$, so B starts higher and falls faster.
Function A has $x$: $2, 4, 6$ and $y$: $11, 17, 23$. Function B starts at $4$ and increases by $2.5$ for each unit. Find A's change in $y$.
$17 - 11 = 6$
Use two neighboring rows.
Find A's change in $x$.
$4 - 2 = 2$
This table steps by $2$, not $1$.
Divide to get A's rate.
$m_A = 6 \div 2 = 3$
The rate is per unit of $x$.
Walk back from $x = 2$ to $x = 0$.
$b_A = 11 - 2 \times 3 = 5$
Each unit back takes $3$ off $y$, and $x = 0$ is $2$ units back.
Write both as equations.
$A: y = 3x + 5, \qquad B: y = 2.5x + 4$
"Starts at" is the initial value and "for each unit" is the rate.
Test A's equation on the last row.
$3 \times 6 + 5 = 23$
It matches the table, so the rate and intercept are right.
Compare the outputs at $x = 10$.
$A: 3 \times 10 + 5 = 35, \qquad B: 2.5 \times 10 + 4 = 29$
Putting the same input into both shows which is larger there.
State the comparison.
$3 > 2.5 \text{ and } 5 > 4$
A has both the larger rate and the larger initial value, so A stays above B for every $x \ge 0$.
Read the equation's rate.
$m_A = 5$
The number multiplying $x$.
Find the table's rate.
$m_B = \dfrac{15 - 7}{3 - 1} = \dfrac{8}{2} = 4$
Change in $y$ over change in $x$.
Walk the table back to $x = 0$.
Compare the two rates and the two starting values.
Function A is $y = 8x + 9$. Function B is given by the table below. Which function has the greater rate of change? $\begin{array}{c|ccc} x & 0 & 1 & 2 \\ \hline y & 4 & 6 & 8 \end{array}$
Function A has the graph through $(0, 10)$ and $(4, 46)$. Function B is $y = 2x + 24$. Complete the worked solution to compare their rates of change.
Find the change in $y$ for Function A.
$46 - 10 =$ u
The rise between the two points on A's graph.
Divide by the change in $x$ to get A's rate.
$m_A =$ a
The run between the points is $4$.
Read B's rate from its equation.
$m_B =$ b
In $y = mx + b$ the rate is the number multiplying $x$.
The graph of a linear function passes through the points $(2, 1)$ and $(5, -14)$. What is its rate of change?
The rate of change is answer.
A linear function is given by this table. Write its equation. $\begin{array}{c|cccc} x & 0 & 2 & 4 & 6 \\ \hline y & 6 & 22 & 38 & 54 \end{array}$
y = mx + b
A linear function is given by this table. What is its $y$-intercept? $\begin{array}{c|ccc} x & 3 & 5 & 7 \\ \hline y & 3 & 7 & 11 \end{array}$
The y-intercept is answer.
Tank A holds $g = 526 - 8t$ gallons after $t$ minutes of draining. Tank B holds $456$ gallons at the start and $404$ gallons after $4$ minutes, draining steadily. How many more gallons per minute drain from Tank B than from Tank A?
Answer:
A copy shop's price list says: $50$ pages cost $372$ cents and $100$ pages cost $622$ cents. A second shop charges $C = 13p + 186$ cents for $p$ pages. Which shop charges more for each extra page?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A linear function is given by this table. Write its equation. $\begin{array}{c|cccc} x & 0 & 2 & 4 & 6 \\ \hline y & 8 & 22 & 36 & 50 \end{array}$
y = mx + b
You can compare functions given in different forms. Without looking: how do you find the rate of change from a table, and from a graph?
16. Your turn: compare $y = 5x - 2$ with the table $x$: $1, 3$ and $y$: $7, 15$, step 3
$b_B = 7 - 1 \times 4 = 3$
One unit back from $x = 1$.
16. Your turn: compare $y = 5x - 2$ with the table $x$: $1, 3$ and $y$: $7, 15$, step 4
$5 > 4, \qquad -2 < 3$
The equation grows faster; the table starts higher.