Topics
7. Quadratic Functions
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United States of AmericaFL
Grade 912

7.01 Quadratic relationships

Lesson
Practice
Lesson

Concept summary

A quadratic relationship is any relationship where the change in output values increases or decreases by a non-zero constant value for each consistent change in x. We can identify a quadratic relationship or function from its equation, a table of values, and by the shape of its graph.

Standard form of a quadratic equation

A form where the quadratic function is expressed as separate polynomial terms, written in descending order of exponents as \\f\left(x\right) = ax^2 + bx + c.

To determine the value by which a quadratic relationship increases or decreases we look at the first difference, which is the difference between consecutive y-values, and then identify the difference between consecutive first differences, known as the second difference. If the second difference is a non-zero constant, we have a quadratic relationship.

Consider a table of values for y=x^2.

x-3-2-10123
f\left( x \right)9410149

We can see the first differences are: -5, -3, -1, +1, +3, +5.

Notice that these values are increasing by 2 each time. This means the second differences have a constant value of 2.

We can draw the graph of y=x^2 and see the general shape of all quadratic relationships. The curve that is formed is known as a parabola.

-4
-3
-2
-1
1
2
3
4
x
-1
1
2
3
4
5
6
7
8
9
10
y
  • This graph shows a quadratic relationship.
  • The graph is symmetrical about a vertical line. In this case x=0.
  • The input, x, can be any value (positive or negative), and the graph continues endlessly to the left and right.
  • The output, y, has a minimum or maximum value. In this case a minimum value of y=0.

Worked examples

Example 1

Functions f\left(x\right), g\left(x\right) and h\left(x\right) are shown below using different representations.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
x-3-2-10123
g\left( x \right)41.5-1-3.5-6-8.5-11

h(x)=2^x

a

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

Approach

We want to examine the graph and determine how the y-values are changing for a consistent change in x.

In a linear relationship, the y-value changes at a constant rate.

In a quadratic relationship, the increase/decrease in the output increases/decreases by a constant value for each consistent change in x. The graph of a quadratic relationship is symmetrical and y has a minimum or maximum value.

In an exponential relationship, the y-value changes by a constant percent.

Solution

The graph has a maximum value of y=4 and is symmetrical, so therefore the graph is quadratic.

b

Identify if g(x) is linear, quadratic, or exponential.

Approach

We want to determine if the table of values has constant first differences (linear) or non-zero constant second differences (quadratic), or if there is a constant percent rate of change (exponential).

Solution

The y-values are decreasing by a constant value of 2.5 for each consecutive increase in x by 1. Therefore, g(x) is linear.

c

Identify if h(x) is linear, quadratic, or exponential.

Approach

A linear equation contains a term of degree 1, but no higher degree.

A quadratic equation contains a term of degree 2, but no higher degree.

An exponential equation has the variable in the exponent.

Solution

The function h(x) has a variable in the exponent, so the function is exponential.

Example 2

Complete the table for the following quadratic function:

x-3-2-1012345
f\left( x \right)0-3-4-35

Approach

We first want to determine if the minimum or maximum value of the quadratic function is shown in the table. If it is, we can use the property of symmetry to help complete any missing values.

To find any remaining values, we want to find the first and second differences and use these to complete the table.

Solution

In this case since f(-1) \text{ and }f(1) are equal, the minimum value occurs at the x-value directly between -1 and 1. The minimum value occurs at \left(0, -4 \right).

Since the quadratic is symmetric about this point, we can use this to find other points. We are given the point \left(-2, 0 \right) so using symmetry we know \left (2, 0 \right) must also be on the function. In other words, when we move 2 units to the left of x=0, the output is 0 so when we move 2 units to the right of x=0 the output must also be 0. For the same reason we know when x=-3, the output is 5.

x-3-2-1012345
f\left( x \right)50-3-4-305

To find the last two values we want to find the first and second differences. Comparing the output values we have found so far, we can see the first differences are: -5, -3, -1, +3, +5, \cdots. This means the second difference is +2.

From this information, we can find the remaining values. When x=4, the output will be 5+2=7 more than when x=3, and the value for x=5 will be 7+2=9 more than this value.

x-3-2-1012345
f\left( x \right)50-3-4-3051221

Reflection

The minimum or maximum will not necessarily be shown in the table, but it is helpful if it is shown due to the symmetry of quadratic relationships.

We could have also noticed that this table follows a similar pattern to the parent quadratic function, f(x)=x^2, but is translated down 4 units. If we noticed this, we could use the rule to complete the missing values.

Outcomes

MA.912.F.1.1

Given an equation or graph that defines a function, determine the function type. Given an input-output table, determine a function type that could represent it.

MA.912.F.1.8

Determine whether a linear, quadratic or exponential function best models a given real-world situation.

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