Rates of Change in Linear and Quadratic Functions - AP PreCalculus Study Guide

The Core Idea: Rates of Change in Linear and Quadratic Functions

This topic explores how we measure the change in a function's output relative to the change in its input over a specific interval. This measurement is called the average rate of change. It provides a single value that describes the function's general behavior across that span. For any function, we can calculate this value using a defined formula, whether we are given an algebraic rule or a table of values.

The core distinction this topic highlights is the fundamental difference in the nature of change between linear and quadratic functions. Linear functions are defined by their constant rate of change; no matter which interval you choose, the average rate of change remains the same. In contrast, quadratic functions have an average rate of change that is not constant. As you move along the parabola, the steepness changes. However, this change is not random; the average rates of change of a quadratic function follow a predictable linear pattern, a key characteristic that helps us identify and understand quadratic behavior.

Key Formula: Average Rate of Change

The average rate of change of a function $f(x)$ over a closed interval $[a,b]$ is the ratio of the change in the function's output values (the "rise") to the change in the input values (the "run").

The formula is:

$$\text{Average Rate of Change}=\frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b-a}$$

Where:

• $f(b)$ is the value of the function at the end of the interval.

• $f(a)$ is the value of the function at the beginning of the interval.

• $b-a$ is the length of the interval.

This formula calculates the slope of the secant line connecting the two points $(a,f(a))$ and $(b,f(b))$ on the graph of the function.

Understanding the Nature of Rates of Change

A crucial skill in precalculus is not just calculating a value, but understanding what that value tells you about the underlying function. The behavior of the average rate of change is a primary way to distinguish between function families.

Linear Functions:

For a linear function, such as $f(x)=mx+c$, the average rate of change is always constant and is equal to the slope, $m$.

Let's calculate the average rate of change over any interval $[a,b]$:

$$\frac{f(b)-f(a)}{b-a}=\frac{(mb+c)-(ma+c)}{b-a}=\frac{mb-ma}{b-a}=\frac{m(b-a)}{b-a}=m$$

As you can see, the result is always $m$, regardless of the interval $[a,b]$. If you calculate the average rate of change from a table of values for a linear function, you will get the same number every time.

Quadratic Functions:

For a quadratic function, such as $g(x)=a{x}^2+bx+c$, the average rate of change is not constant. It depends on the specific interval you choose. However, these rates of change exhibit a distinct linear pattern. If you calculate the average rates of change over consecutive intervals of equal length, you will find that the differences between those rates are constant.

A special point of interest for a quadratic function is its vertex. The vertex of a parabola represents the point where the function changes from increasing to decreasing, or vice versa. At this turning point, the rate of change is momentarily zero. While the concept of an "instantaneous" rate of change is reserved for calculus, we can observe this property by calculating the average rate of change on an interval that is symmetric around the vertex. For any interval $[x_v-c,x_v+c]$ centered on the vertex's x-coordinate $x_v$, the average rate of change will be zero because $f(x_v-c)=f(x_v+c)$, making the numerator $f(b)-f(a)$ equal to zero.

Core Concepts & Rules

• Definition: The average rate of change of a function $f$ over an interval $[a,b]$ is given by the formula $\frac{f(b)-f(a)}{b-a}$.

• Calculation: This rate can be calculated directly from a function's algebraic formula or from a given table of input-output values.

• Linear Function Property: A function is linear if and only if its average rate of change is constant for all intervals.

• Quadratic Function Property: The average rates of change of a quadratic function are not constant.

• Quadratic Rate Pattern: The average rates of change of a quadratic function, when calculated over consecutive intervals of equal length, will always form a linear pattern (i.e., they increase or decrease by a constant amount).

• The Vertex: The vertex of a parabola is the point where the function's rate of change is zero. This can be seen by calculating the average rate of change over an interval that is symmetric about the vertex, which will yield a result of 0.

Step-by-Step Example 1: Calculating from a Function's Formula

Problem: Find the average rate of change of the quadratic function $f(x)=2x^2-3x+1$ on the interval $[1,4]$.

Step 1: Identify the interval endpoints.

Here, $a=1$ and $b=4$.

Step 2: Evaluate the function at the endpoint, $f(b)$.

We need to calculate $f(4)$.

$$f(4)=2(4{)}^2-3(4)+1$$

$$f(4)=2(16)-12+1$$

$$f(4)=32-12+1=21$$

Step 3: Evaluate the function at the start point, $f(a)$.

We need to calculate $f(1)$.

$$f(1)=2(1{)}^2-3(1)+1$$

$$f(1)=2(1)-3+1$$

$$f(1)=2-3+1=0$$

Step 4: Substitute the values into the average rate of change formula.

The formula is $\frac{f(b)-f(a)}{b-a}$.

$$\text{Average Rate of Change}=\frac{f(4)-f(1)}{4-1}$$

$$\text{Average Rate of Change}=\frac{21-0}{3}$$

Step 5: Calculate the final result.

$$\text{Average Rate of Change}=\frac{21}{3}=7$$

Conclusion: The average rate of change of $f(x)=2x^2-3x+1$ on the interval $[1,4]$ is 7.

Step-by-Step Example 2: Analyzing Rates of Change from a Table

Problem: A function $g(x)$ is represented by the table of values below.

(a) Calculate the average rates of change for $g(x)$ over the consecutive intervals $[0,1]$, $[1,2]$, $[2,3]$, and $[3,4]$.

(b) Based on the results from part (a), what type of function could $g(x)$ be? Justify your answer.

(c) Identify the interval where the average rate of change is 0 and explain what this implies about the function's graph.

Solution:

(a) Calculate the average rates of change.

• Interval [0, 1]:

  $$\frac{g(1)-g(0)}{1-0}=\frac{4-10}{1}=-6$$

• Interval [1, 2]:

  $$\frac{g(2)-g(1)}{2-1}=\frac{0-4}{1}=-4$$

• Interval [2, 3]:

  $$\frac{g(3)-g(2)}{3-2}=\frac{-2-0}{1}=-2$$

• Interval [3, 4]:

  $$\frac{g(4)-g(3)}{4-3}=\frac{0-(-2)}{1}=2$$

(b) Determine the function type.

The average rates of change are -6, -4, -2, and 2.

• The rates of change are not constant, so the function is not linear.

• Let's examine the pattern in the rates of change. The difference between consecutive rates is:

  • $(-4)-(-6)=2$

  • $(-2)-(-4)=2$

  • $2-(-2)=4$ (Wait, let me recheck my calculation for [3,4]... Ah, $g(4)=0$ and $g(3)=-2$. So $0-(-2)=+2$. The rate is 2. Let's recheck the pattern. -6, -4, -2, 2. The difference is +2, +2, +4. This is not a linear pattern. Let me re-create the table to ensure it represents a quadratic. A good quadratic would be $g(x)=2x^2-8x+10$. Let's check the values.

  $g(-1)=2+8+10=20$ (not 18)

  $g(0)=10$ (match)

  $g(1)=2-8+10=4$ (match)

  $g(2)=2(4)-16+10=8-16+10=2$ (not 0)

  Let's try another one: $g(x)=x^2-5x+4$.

  $g(0)=4$ (no)

  Let's try $g(x)=a{x}^2+bx+c$.

  $g(0)=10⟹c=10$.

  $g(1)=4⟹a+b+10=4⟹a+b=-6$.

  $g(2)=0⟹4a+2b+10=0⟹2a+b=-5$.

  Subtracting the two equations: $(2a+b)-(a+b)=-5-(-6)⟹a=1$.

  Then $1+b=-6⟹b=-7$.

  So the function is $g(x)=x^2-7x+10$.

  Let's check the table with this function:

  $g(-1)=1+7+10=18$ (match)

  $g(0)=10$ (match)

  $g(1)=1-7+10=4$ (match)

  $g(2)=4-14+10=0$ (match)

  $g(3)=9-21+10=-2$ (match)

  $g(4)=16-28+10=-2$ (not 0).

  $g(5)=25-35+10=0$ (not 4).

  The table provided in my scratchpad was flawed. I must create a valid table for a quadratic to demonstrate the principle. Let's use $g(x)=x^2-6x+8$.

  $x=0,g(x)=8$

  $x=1,g(x)=1-6+8=3$

  $x=2,g(x)=4-12+8=0$

  $x=3,g(x)=9-18+8=-1$

  $x=4,g(x)=16-24+8=0$

  $x=5,g(x)=25-30+8=3$

  $x=6,g(x)=36-36+8=8$

  This is a good symmetric table.

Let's restart the example with a correct table.

Problem: A function $g(x)$ is represented by the table of values below.

(a) Calculate the average rates of change for $g(x)$ over the consecutive intervals $[1,2]$, $[2,3]$, $[3,4]$, and $[4,5]$.

(b) Based on the results from part (a), what type of function could $g(x)$ be? Justify your answer.

(c) Calculate the average rate of change on the interval $[2,4]$ and explain what this implies about the function's graph.

Solution:

(a) Calculate the average rates of change.

• Interval [1, 2]:

  $$\frac{g(2)-g(1)}{2-1}=\frac{0-3}{1}=-3$$

• Interval [2, 3]:

  $$\frac{g(3)-g(2)}{3-2}=\frac{-1-0}{1}=-1$$

• Interval [3, 4]:

  $$\frac{g(4)-g(3)}{4-3}=\frac{0-(-1)}{1}=1$$

• Interval [4, 5]:

  $$\frac{g(5)-g(4)}{5-4}=\frac{3-0}{1}=3$$

(b) Determine the function type.

The average rates of change are -3, -1, 1, and 3.

• The rates of change are not constant, so the function is not linear.

• Let's examine the pattern in the rates of change. The difference between consecutive rates is:

  • $(-1)-(-3)=2$

  • $1-(-1)=2$

  • $3-1=2$

• Since the average rates of change increase by a constant amount (2) for each consecutive interval of length 1, they form a linear pattern. This is a key characteristic of a quadratic function.

(c) Analyze the interval [2, 4].

• Calculate the average rate of change on [2, 4]:

  $$\frac{g(4)-g(2)}{4-2}=\frac{0-0}{2}=\frac{0}{2}=0$$

• Implication: An average rate of change of 0 on the interval $[2,4]$ implies that the function's starting value, $g(2)$, is the same as its ending value, $g(4)$. Because the function is quadratic, this symmetry implies that the vertex of the parabola must lie within this interval, specifically at the midpoint $x=3$. The vertex is the point where the function's rate of change is 0.

Using Your Calculator

While the average rate of change is a straightforward concept to calculate by hand, a graphing calculator (like a TI-84) can speed up the process, especially with more complex functions, and help prevent arithmetic errors. The main use is for quick and accurate function evaluation.

Problem: Find the average rate of change of $f(x)=\frac{x^3-\sin (x)}{x^2+1}$ on the interval $[2,5]‘.\ast \ast StepsonaTI-84StyleCalculator:\ast \ast 1.\ast \ast EntertheFunction:\ast \ast -Pressthe‘Y=‘button.-In‘Y1‘,typethefunction.Becarefulwithparentheses:$(X^3 - sin(X)) / (X^2 + 1)$. Make sure your calculator is in RADIAN mode for trigonometric functions.

2. Go to the Home Screen:

   • Press 2nd then MODE (QUIT) to return to the main calculation screen. 3. **Use the Formula with Y-VARS:** - Type an open parenthesis $(`.

   • Press VARS, go to the Y-VARS menu, select 1:Function..., and then 1:Y1.

   • Type (5)) to evaluate $f(5)$. Your screen should show Y1(5).

   • Type -.

   • Press VARS, go to Y-VARS, select 1:Function..., and then 1:Y1.

   • Type (2))to evaluate $f(2). Your screen should now show Y1(5) - Y1(2)`.

   • Close the numerator with a parenthesis $)$.

   • Type / for division.

   • Type (5 - 2).

   • Your full entry should look like: (Y1(5) - Y1(2)) / (5 - 2)

3. Calculate:

   • Press ENTER. The calculator will compute the result.

   • Result: $(\mathrm{4.784...}-\mathrm{1.418...})/3\approx 1.122$

This method is highly efficient and reduces the chance of errors in substitution and calculation.

AP Exam Quick Hit

Common Question Types

• Direct Calculation from a Function: You will be given a function, $f(x)$, and an interval, $[a,b]$, and asked to find the average rate of change.

  • Example: "What is the average rate of change of the function $h(t)=4t^2-7t$ on the interval $[1,3]$?"

• Analysis of a Table of Values: You will be given a table of values for a function and asked to calculate and interpret the average rates of change to determine if the function could be linear or quadratic.

  • Example: "The table below shows values for a function $f$. By analyzing the average rates of change over consecutive unit intervals, determine if $f$ could be a linear function, a quadratic function, or neither."

• Graphical Interpretation: You may be shown a graph of a function and asked to identify an interval where the average rate of change is positive, negative, or zero, or to compare the average rate of change over two different intervals.

  • Example: "For the quadratic function graphed above, on which of the following intervals is the average rate of change equal to 0? (A) [-2, 0] (B) [0, 2] (C) [-1, 3] (D) [1, 3]" (The correct answer would be the interval symmetric about the vertex).

Common Mistakes

• Incorrect Formula Order: Swapping the numerator and denominator, calculating $\frac{b-a}{f(b)-f(a)}$ instead of the correct $\frac{f(b)-f(a)}{b-a}$. Remember that rate of change is always "change in output" over "change in input."

• Sign Errors in Subtraction: Forgetting to distribute the negative sign when calculating $f(b)-f(a)$, especially when $f(a)$ is a negative value or a multi-term expression. For example, if $f(x)=x^2-5$ and you are calculating $f(3)-f(1)$, a common error is writing $4-(-4)$ as $4-4$ instead of $4+4$.

• Confusing Average Rate of Change with Slope at a Point: For a quadratic function, the average rate of change over an interval is not the same as the "steepness" at a single point. The average rate of change is the slope of the line between the two endpoints of the interval.

• Misinterpreting the Vertex: Believing the average rate of change is 0 only at the vertex point itself. The average rate of change is calculated over an interval. The average rate of change is 0 over any interval $[a,b]$ where $f(a)=f(b)$. For a parabola, this occurs on any interval that is symmetric around the vertex's axis of symmetry.