Topics
7. Quadratic Functions
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United States of AmericaTennessee
Algebra 1

7.07 Linear, quadratic, and exponential models

Lesson
Practice
Lesson

Concept summary

Linear, quadratic and exponential functions differ in how the output of the function changes with regards to the input x.

We can use three things to determine the change in output: the first difference, the second difference, and the common ratio.

xf(x)\text{first difference}
-2-4
-1-2-2-(-4)=2
000-(-2)=2
122-0=2
244-2=2
366-4=2

The first difference is the difference between two consecutive outputs.

If the first difference is constant, then the function is linear.

xg(x)\text{first} \\ \text{difference}\text{second} \\ \text{difference}
-2-4
-1-22
0244-2=2
1866-4=2
21688-6=2
3261010-8=2

The second difference is the difference between two consecutive first differences.

If the second difference is constant and the first difference is not, then the function is quadratic.

xf(x)\text{common ratio}
-2\dfrac{1}{4}
-1\dfrac{1}{2}\dfrac{1}{2}\div \dfrac{1}{4}=2
011\div \dfrac{1}{2}=2
122\div 1=2
244\div 2=2
388\div 4=2

If neither the first nor second difference is constant, then we look at the common ratio. If the common ratio is constant then the function is exponential.

We can also identify a function by the shape of its graph:

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
  • f\left(x\right) is a quadratic function.
  • g\left(x\right) is a linear function
  • h\left(x\right) is an exponential function

Worked examples

Example 1

Identify if the function f(x) is linear, quadratic or exponential.

x-2-10123
f\left( x \right)131-311333

Approach

Construct a table to find the common ratio and first and second difference.

  • If the common ratio is constant for all x, then f(x) is exponential.

  • If the first difference is constant for all x, then f(x) is linear.

  • If the second difference is constant for all x, and the first difference is not, then f(x) is quadratic.

Solution

We get the following table of values:

xf(x)\text{Common} \\ \text{ ratio}\text{First} \\ \text{ difference}\text{Second} \\ \text{ difference}
-213
-11\dfrac{1}{13}-12
0-3-3-48
11-\dfrac{1}{3}48
21313128
333\dfrac{13}{33}208

The common ratio and the first difference are not constant, so the function is not exponential or linear.

The second difference is constant, so the function is quadratic.

Reflection

If the common ratio and the first and second differences are not constant, then it cannot be linear, quadratic or exponential.

Example 2

Find the average rate of change of the function f(x) between x = 2 and x = 6.

-1
1
2
3
4
5
6
7
8
x
1
2
3
4
5
6
7
8
f(x)

Approach

Find the values of f(2) and f(6) using the graph.

Divide the difference between the y-values by the change in x. In this case getting the expression

\frac{f(6)-f(2)}{6-2}

Evalute the expression to find the average rate of change.

Solution

\displaystyle \text{Average rate of change}\displaystyle =\displaystyle \frac{f(6)-f(2)}{6-2}
\displaystyle {}\displaystyle =\displaystyle \frac{5-1}{6-2}Substitute the values in for f(6) and f(2)
\displaystyle {}\displaystyle =\displaystyle \frac{4}{4}Evaluate the numerator and denominator
\displaystyle {}\displaystyle =\displaystyle 1Cancel the common factors in the fraction

Reflection

The average rate of change is the slope of the straight line between the two points.

Example 3

Would a linear, quadratic or exponential function best model the number of pieces a cake is cut into, if each piece of cake is cut in half every minute?

Assume you start with a whole cake.

Approach

Construct a table with the number of pieces of cake.

Construct a first differences, second differences and the common ratios table of consecutive terms to identify whether there are any which are constant.

  • If the common ratio is constant for all x, then f(x) is exponential.

  • If the first difference is constant for all x, then f(x) is linear.

  • If the second difference is constant for all x, and the first difference is not, then f(x) is quadratic.

Determine whether the situation is best described be a linear, quadratic or exponential function.

Solution

We get the following table of values:

\text{Minutes} \\ \text{ passed}\text{Pieces} \\ \text{ of cake}\text{Common} \\ \text{ ratio}\text{First} \\ \text{ difference}\text{Second} \\ \text{ difference}
01
1221
24221
38242
416284
5322168

The common ratio is constant for all values, therefore the best function to model this scenario is an exponential function.

Example 4

Compare the key features of quadratic and exponential functions.

Approach

Think about how you would describe each feature for each function type. It can help to sketch a few examples of each function type to visualize these features.

Solution

The table below is an example of how we could compare the key features of these two types of functions.

\text{Quadratic}\text{Exponential}
\text{number of } \\ x-\text{intercept(s)}0, 1 \text{ or } 20 \text{ or } 1
y-\text{intercept}\text{Yes}\text{Yes}
\text{Min/Max point}\text{Yes}\text{No}
\text{Domain}\text{All reals}\text{All reals}
\text{Range}\text{Limited by } \text{min/max}\text{Limited by }\text{asymptote}
\text{Asymptote}\text{No}\text{Yes}
\text{End behavior}\text{Same }\left(\text{both }\infty\text{ or }-\infty\right)\text{Different }\left(\text{Asymptote and }\infty\text{ or }-\infty\right)
\text{Leading term}ax^2ab^x

Reflection

The example solution above is not the only way to fill the table. You can be more specific, provide examples, or sketch small diagrams to help you understand each feature.

Outcomes

A1.N.Q.A.1

Use units as a way to understand real-world problems.*

A1.N.Q.A.1.A

Choose and interpret the scale and the origin in graphs and data displays.*

A1.N.Q.A.1.C

Define and justify appropriate quantities within a context for the purpose of modeling.*

A1.N.Q.A.1.D

Choose an appropriate level of accuracy when reporting quantities.*

A1.A.CED.A.2

Create equations in two variables to represent relationships between quantities and use them to solve problems in a real-world context. Graph equations with two variables on coordinate axes with labels and scales, and use the graphs to make predictions.*

A1.A.CED.A.3

Create individual and systems of equations and/or inequalities to represent constraints in a contextual situation, and interpret solutions as viable or non-viable.*

A1.A.REI.D.5

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A1.A.REI.D.6

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x). Find approximate solutions by graphing the functions or making a table of values, using technology when appropriate.*

A1.F.IF.B.5

Relate the domain of a function to its graph and, where applicable, to the context of the function it models. *

A1.F.IF.C.8.A

Rewrite quadratic functions to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a real-world context.

A1.F.IF.C.9

Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.*

A1.F.IF.C.9.A

Compare properties of two different functions. Functions may be of different types and/or represented in different ways.

A1.F.LE.A.1

Distinguish between situations that can be modeled with linear functions and with exponential functions.*

A1.MP1

Make sense of problems and persevere in solving them.

A1.MP2

Reason abstractly and quantitatively.

A1.MP3

Construct viable arguments and critique the reasoning of others.

A1.MP4

Model with mathematics.

A1.MP5

Use appropriate tools strategically.

A1.MP6

Attend to precision.

A1.MP7

Look for and make use of structure.

A1.MP8

Look for and express regularity in repeated reasoning.

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