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2. Quadratic Functions
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Integrated Math 2

2.01 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
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x
-1
1
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4
5
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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
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x
-4
-3
-2
-1
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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.

Example 3

The following graph of the function, y=f(x), represents the path of a ball kicked into the air.

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x\left(\text{secs}\right)
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y\left(\text{ft}\right)
a

Interpret the real-world meaning of the highest point of the graph. Be as specific as possible.

Approach

We first want to determine where the highest point of the graph is. Once we identify that point, we can then interpret the real-world meaning based on the units and context provided.

Solution

The highest point of the graph is located at (5, 9). If we consider the units provided, we know that this point in a real-life context represents the ball being at a height of 9 feet after 5 seconds has passed.

Since this is the highest point of the curve, we also know that the height of the ball will not be greater than 9 feet. So the highest point of the graph represents the highest height that the ball will reach in feet, regardless of time.

b

Interpret the real-world meaning of the x-intercepts. Be as specific as possible.

Approach

We first want to determine where the x-intercepts of the graph are. Once we identify the points, we can then interpret the real-world meaning based on the units and context provided.

Solution

The x-intercepts of the graph are found by identifying where the graph crosses the x-axis. The graph crosses the x-axis at (2, 0) and (8, 0).

If we consider the units provided, these two points represent when the height of the ball is 0 feet after 2 and 8 seconds respectively. Since the y-value represents the height of the ball, these points correspond to a height of 0 feet, which is when the ball is on the ground.

Since this is happening twice, at 2 and 8 seconds, the x-intercepts represent when the ball is kicked (at a time of 2 seconds) and when it comes back down to the ground (at 8 seconds).

Outcomes

M2.N.Q.A.1

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

M2.N.Q.A.1.A

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

M2.A.SSE.A.1

Interpret expressions that represent a quantity in terms of its context.*

M2.A.SSE.A.1.A

Interpret parts of an expression, such as terms, factors, and coefficients.

M2.A.SSE.A.1.B

Interpret complicated expressions by viewing one or more of their parts as a single entity.

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP2

Reason abstractly and quantitatively.

M2.MP3

Construct viable arguments and critique the reasoning of others.

M2.MP4

Model with mathematics.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

M2.MP8

Look for and express regularity in repeated reasoning.

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