Learning activity 2.6: Sketching and analyzing graphs of quadratic functions

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At this point in the course, you are familiar with three forms of quadratic equations. Let’s now analyze a real-world problem using the skills we have learned so far!

In this learning activity, we will introduce the key features required to graph and analyze quadratic functions. Having a deep understanding of these properties plays an important part in how you go about extracting information arising from real life situations.

Parabolas have an axis of symmetry. This is a line that would make the graph symmetrical on either side.

Explore this!

To learn more about this you may explore the following video, until 1 minute and 13 seconds that explains an axis of symmetry.

Now let’s review the features of a parabola by pressing on each of the following tabs.

Direction of opening

A parabola opens either upwards or downwards.

If the parabola opens upwards, the graph has a minimum value.

If the parabola opens downwards, the graph has a maximum value.

Vertex

The highest or lowest point on a parabola (depending on which way it opens).

The vertex of a parabola that opens upwards is at the bottom. The vertex of a parabola that opens downwards is at the top.

On the graph beside, the vertex is (–1, –3).

Maximum / minimum value

The highest or lowest y-value on the graph (in other words, this is the y-value of the vertex).

Since the graph beside opens upwards, it has a minimum value of y = –3.

x-intercept(s)

The points where the parabola crosses (intersects) the x-axis. The x-intercepts occur where y = 0.

On the graph beside, there are two x-intercepts, at 0.75 and –2.75. But some parabolas don't cross the x-axis so they don't have x-intercepts. And others have just one x-intercept (when the vertex is at y = 0).

y-intercept

The point where the parabola crosses or intersects the y-axis. The y-intercept occurs where x = 0.

On the graph beside, the y-intercept is –2.

Axis of symmetry

The axis of symmetry is the line that runs down the centre of the parabola and goes through the vertex. It's also called a 'mirror line' because the graph is symmetrical on either side of it.

On the following graph, the axis of symmetry is the vertical line x = –1.

Notebook

Answer the following questions in your notebook. Be sure to check your work with the suggested answers provided.

What point does the axis of symmetry always go through?

The vertex.

How does the $x$ -value of the vertex relate to the $x$ -intercepts?

It is the midpoint between the $x$ -intercepts. This would also be called the average or mean of the $x$ -intercepts.

Sketching graphs from factored form

You learned in the Minds On section that the $x$ value of the vertex, the value of the axis of symmetry, is directly in the middle of the $x$ -intercepts.

You can use the axis of symmetry value to find the $y$ value of the vertex, called the optimal value.

Use the example: the area of a fenced backyard is represented by the equation $A = - (2 x - 3) (2 x - 9)$ , where $A$ is the area and $x$ is an unknown measure contributing to the length and width. Determine the maximum area that can be fenced in given the dimensions.

Step 1: Determine the $x$ -intercepts

You may want to review how to find the $x$ -intercepts from factored form from earlier in the course.

$\begin{array}{l} A = - (2 x - 3) (2 x - 9) \\ 0 = - (2 x - 3) (2 x - 9) \\ 0 = 2 x - 3 0 = 2 x - 9 \\ 3 = 2 x 9 = 2 x \\ \frac{3}{2} = x \frac{9}{2} = x \\ (\frac{3}{2}, 0) and (\frac{9}{2}, 0) \end{array}$

Step 2: Determine the $x$ value of the vertex (the axis of symmetry value).

Note that the $x$ -value of the vertex is the average, or middle, of the $x$ -intercepts.

$x = \frac{\frac{3}{2} + \frac{9}{2}}{2}$

$x = \frac{6}{2}$

$x = 3$

The $x$ value is $3$ .

Step 3: Substitute the $x$ -value of the vertex into the original equation to find the $y$ -value of the vertex.

Note that in the following example, the ‘ $y$ ’ values are ‘ $A$ ’ values.

$A = - (2 x - 3) (2 x - 9)$

$A = - (2 (3) - 3) (2 (3) - 9)$

$A = - (6 - 3) (6 - 9)$

$A = - (3) (- 3)$

$A = 9$

Therefore the maximum area that can be obtained is 9.

Step 4: Sketch if necessary.

You must plot your $x$ -intercepts and the vertex.

Example 1

Complete an analysis for the quadratic function $y = 0.5 (x - 8) (x + 1)$ .

Determine the $x$ -intercepts.

To find the $x$ -intercepts, let $y = 0$ .

$0 = 0.5 (x - 8) (x + 1)$

$x - 8 = 0$ or $x + 1 = 0$

$x = 8$ or $x = - 1$

Determine the coordinates of the vertex.

To find the vertex, first find the mid-value.

$x = \frac{8 + (- 1)}{2}$

$x = \frac{7}{2}$

$x = 3.5$

To find the y-coordinate of the vertex, substitute $x = 3.5$ into the equation.

$y = 0.5 ((3.5) - 8) ((3.5) + 1)$

$y = 0.5 (- 4.5) (4.5)$

$y = - 10.125$

The vertex is $(3.5, - 10.125)$ .

Sketch the graph of the function.

To graph the parabola, plot the $x$ -intercepts and the vertex. Join the points with a smooth curve, as shown in the following graph.

State the intervals where the function is positive or negative.

From the graph, you can notice that the y-values are above the $x$ -axis for $x < - 1$ and $x > 8$ . Therefore, the function is positive for $x < - 1$ and $x > 8$ .

From the graph you can notice that the y-values are below the $x$ -axis for $- 1 < x < 8$ . Therefore, the function is negative for $- 1 < x < 8$ .

State the intervals where the function is increasing (moving from left to right, the $y$ -values are increasing) or decreasing (moving from left to right, the $y$ -values are decreasing).

From the graph, you can notice that the parabola is increasing when $x > 3.5$ .

From the graph, you can notice that the parabola is decreasing when $x < 3.5$ .

Example 2

Complete an analysis for the quadratic function $y = - 6 (x + 1) (x - 3)$ .

Determine the $x$ -intercepts.

To find the $x$ -intercepts, let $y = 0$ .

$0 = - 6 (x + 1) (x - 3)$

$x + 1 = 0$ or $x - 3 = 0$

$x = - 1$ or $x = 3$

Determine the coordinates of the vertex.

To find the vertex, first find the mid-value.

$x = \frac{(- 1) + 3}{2}$

$x = \frac{2}{2}$

$x = 1$

To find the y-coordinate of the vertex, substitute $x = 1$ into the equation.

$y = - 6 ((1) + 1) ((1) - 3)$

$y = - 6 (2) (- 2)$

$y = 24$

The vertex is $(1, 24)$ .

Sketch the graph of the function.

To graph the parabola, plot the $x$ -intercepts and the vertex. Join the points with a smooth curve, as shown in the following graph.

State the intervals where the function is positive or negative.

From the graph, you can notice that the y-values are above the x-axis for $- 1 < x < 3$ . Therefore, the function is positive for $- 1 < x < 3$ .

From the graph, you can notice that the y-values are below the x-axis for $x < - 1$ and $x > 3$ . Therefore, the function is negative for $x < - 1$ and $x > 3$ .

State the intervals where the function is increasing or decreasing.

From the graph, you can notice that the parabola is increasing (moving from left to right, the $y$ -values are going up) when $x < 1$ .

From the graph, you can notice that the parabola is decreasing (moving from left to right, the $y$ -values are going down) when $x > 1$ .

Example 3

Complete an analysis for the quadratic function $y = \frac{3}{4} (x - 4) (x + 4)$ .

Determine the $x$ -intercepts.

To find the $x$ -intercepts, let $y = 0$ .

$0 = \frac{3}{4} (x - 4) (x + 4)$

$x - 4 = 0$ or $x + 4 = 0$

$x = 4$ or $x = - 4$

Determine the coordinates of the vertex.

To find the vertex, first find the mid-value.

$x = \frac{(- 4) + 4}{2}$

$x = \frac{0}{2}$

$x = 0$

To find the y-coordinate of the vertex, substitute x=0 into the equation.

$y = \frac{3}{4} ((0) - 4) ((0) + 4)$

$y = \frac{3}{4} (- 4) (4)$

$y = - 12$

The vertex is $(0, - 12)$ .

Sketch the graph of the function.

To graph the parabola, plot the $x$ -intercepts and the vertex. Join the points with a smooth curve, as shown in the following graph.

State the intervals where the function is positive or negative.

From the graph, you can notice that the y-values are above the $x$ -axis for $x < - 4$ and $x > 4$ . Therefore, the function is positive for $x < - 4$ and $x > 4$ .

From the graph, you can notice that the y-values are below the $x$ -axis for $- 4 < x < 4$ . Therefore, the function is negative for $- 4 < x < 4$ .

State the intervals where the function is increasing or decreasing.

From the graph, you can notice that the parabola is increasing (going up) when $x > 0$ .

From the graph, you can notice that the parabola is decreasing (going down) when $x < 0$ .

Information from equations of a quadratic

Earlier in the course you were asked to summarize each form of the quadratic in your math journal. It may be helpful to refer to that and add to it if necessary. The following is one way to summarize what you now know about the three forms of the quadratic equation.

Quadratic equation form:	Vertex form	Standard form	Factored form
Equation	$y = a (x - h)^{2} + k$	$y = a x^{2} + b x + c$	$y = a (x - r) (x - s)$
Vertex	$(h, k)$	Convert to vertex form the use $(h, k)$	The mid-value and the equation of the axis of symmetry are easily determined from the $x$ -intercepts. The vertex is found using the mid-value.
Roots or $x$ -intercepts	Convert to factored form to get roots or $x$ -intercepts. If the equation is not factorable, the quadratic formula is applied	Equation must be converted to factored. If the equation is not factorable, the quadratic formula is applied.	$(r, 0)$ and $(s, 0)$
$y$ -intercept	Convert to standard form to get $y$ -intercept	The c-value determines the $y$ -intercept, which is $(0, c)$	Convert to standard form to get $y$ -intercept
Concave up or down	Convert to standard form or factored form	Positive $a$ -value determines concave up	Positive $a$ -value determines concave up

Notebook

In your notebook, answer the following questions based on the equation given:

1. A ball is thrown and is represented by the equation $h (t) = - \frac{1}{10} (t - 6)^{2} + 4,$ where $h$ is the height of the ball in meters and $t$ is the time in seconds.

a) What form is the equation in?

The equation is in vertex form.

b) What information about the situation can you determine directly from the equation?

The parabola has a maximum point since $a < 1$ so it is concave down.

The vertex is $(6, 4)$ which tells us the maximum height is $4 m$ after $6 s$ .

2. The population growth for a town is represented by the equation $P (x) = 3 x^{2} + 6 x + 80$ , where $P$ is the population in thousands and $x$ is the number of years, where $x = 0$ is the year 2018.

a) What form is the equation in?

Standard form.

b) What information about the situation can you determine directly from the equation?

The parabola has a minimum point since $a > 1$ so it is concave up.

The $y$ intercept is $(0, 80)$ which tells us that in the year 2018, the town has a population of 80,000.

3. A car dealer’s profit is modelled by the equation $P (x) = - (x - 5) (x - 20)$ , where $P$ is the company’s profit in millions and x is the number of cars sold in thousands.

a) What form is the equation in?

Factored form.

b) What information about the situation can you determine directly from the equation?

The parabola has a maximum point since $a < 1$ so it is concave down.

The $x$ -intercepts are $(5, 0)$ and $(20, 0)$ which tells us that the break-even points (when the profit is 0), are after 5,000 and again after 20,000 cars sold.

Graphing with the step method

We have so far learned two methods on graphing a parabola.

Using the vertex and two symmetrical points.
Using the $x$ -intercepts, axis of symmetry, and optimal value.

The third way of graphing given an equation is to use the step pattern of the parabola.

Examine the points on the following graph of $y = x^{2}$ . The points on the right side of the vertex are spaced as follows: from (0,0) move 1 right and 1 up, then move 1 right and 3 up, then move 1 right and 5 up, then move 1 right and 7 up, and so on.

Follow the step method for points on the left of the vertex: from $(0, 0)$ move 1 left and 1 up, then move 1 left and 3 up, then move 1 left and 5 up, then move 1 left and 7 up.

In the table beside, observe that the $y$ -values to the right of the vertex increase by 1, 3, 5, and 7. The same is true for the $y$ -values to the left of the vertex.

$x$	$y$
-4	16
-3	9
-2	4
-1	1
0	0
1	1
2	4
3	9
4	16

Now notice how this pattern works when $a = 2$ for the following graph of $y = 2 x^{2}$ .

Notebook

In your notebook, answer the following questions. Be sure to check your work with the suggested answers provided.

From the vertex $(0, 0)$ , what is the step pattern to the right?

1 right and 2 up, then 1 right and 6 up, then 1 right and 10 up, then 1 right and 14 up, and so on.

From the vertex $(0, 0)$ , what is the step pattern to the left?

1 left and 2 up, then 1 left and 6 up, then 1 left and 10 up, then 1 left and 14 up, and so on.

How does the step pattern for $y = 2 x^{2}$ relate to the step pattern of $y = x^{2}$ ?

This time, the upward increments are $1 (2) = 2$ , $3 (2) = 6$ , $5 (2) = 10$ , $7 (2) = 14$ , and so on. So in general, the increments of the $y$ -values are $1 a$ , $3 a$ , $5 a$ , $7 a$ , and so on.

When a is negative, move down from the vertex instead of up, as shown by the graph of $y = - x^{2}$ .

From the vertex $(0, 0)$ , what is the step pattern to the right?

1 right and 1 down, then 1 right and 3 down, then 1 right and 5 down, then 1 right and 7 down, and so on.

From the vertex $(0, 0)$ , what is the step pattern to the left?

1 left and 1 down, then 1 left and 3 down, then 1 left and 5 down, then 1 left and 7 down, and so on.

The same is true for the graph of $y = - 2 x^{2}$ .

From the vertex $(0, 0)$ , what is the step pattern to the right?

1 right and 2 down, then 1 right and 6 down, then 1 right and 10 down, then 1 right and 14 down, and so on.

From the vertex $(0, 0)$ , what is the step pattern to the left?

1 left and 2 down, then 1 left and 6 down, then 1 left and 10 down, then 1 left and 14 down, and so on.

Graph the following using the step pattern methods and answer the following questions:

$y = - 3 (x + 4)^{2} + 2$

Identify and plot the vertex.

The vertex $(h, k)$ is $(- 4, 2)$

Identify and apply the step pattern on the left and right side of the graph of the vertex.

The step pattern is multiplied by 3 and will go down because a is negative.

1 right, 3 down, 1 right, 9 down, 1 right, 15 down, etc.

1 left, 3 down, 1 left, 9 down, 1 left, 15 down, etc.

The following is what it is like on the graph.

State the intervals of increase and decrease.

From the graph, you can notice the parabola is increasing (going up) for $x < - 4$ .

From the graph, you can notice the parabola is decreasing (going down) for $x > - 4$ .

Self-check

Rate your level of understanding from 1 (I am still confused) to 5 (I fully understand this concept) based on your results from the questions you just completed.

I am able to:

Statement	1	2	3	4	5
Sketch a graph from factored form by identifying the roots and the vertex
Identify key features of a quadratic given the equation
Graph a parabola using the step pattern method

If there are any criteria where you rated your level of understanding a 3 or below, you should review the concepts before moving on to the next learning activity.

If you feel you need more practice visualizing when solve quadratic equations, you may want to use an online graphing tool of your choice to see the relation on a graph to aid in finding the solution.

Math journal

At the end of the course, you will fine-tune 8 entries (two from each unit) from your math journal and submit them as your “Culminating Assessment - Math Journal.” (Opens in new window)

In your math journal, define the following quadratic properties and include all information about each that you think will be helpful. You can do this in an online journal, a written journal, video journal, audio journal, etc. A diagram may also be useful for reviewing later.

vertex
x-intercept
y-intercept
axis of symmetry
optimal value
step pattern
direction of opening
graph a parabola using the step pattern method

Portfolio

Check your understanding

In your notebook, determine the $x$ -intercepts for each of the following equations.

1. In your notebook, represent the equation for the following graph in factored form $y = a (x \pm r) (x \pm s)$ .

2. Represent the equation for the following graph in vertex form $y = a (x - h)^{2} + k$ .

3. Find the equation of the following graph in vertex form. Expand that equation to represent the equation in standard form $y = a x^{2} + b x + c$ .

4. Represent the equation for the following graph in standard form $y = a x^{2} + b x + c$ .

5. Represent the equation for the following graph in factored form $y = a (x \pm r) (x \pm s)$ .

6. Represent the equation for the following graph in factored form $y = a (x \pm r) (x \pm s)$ .

7. Represent the equation for the following graph in factored form $y = a (x \pm r) (x \pm s)$ .

8. Represent the equation for the following graph in vertex form $y = a (x - h)^{2} + k$ .

9. Represent the equation for the following graph in vertex form $y = a (x \pm h)^{2} \pm k$ .

10. Represent the equation for the following graph in vertex form $y = a (x \pm h)^{2} \pm k$ .

Summary of diagnosing transformations and quadratic equations

11. What is the first step you should take when diagnosing transformations?

12. How do you convert a quadratic function into its vertex form?

Submit your portfolio item(s) by pressing the “Go To Portfolio” button.

Review of the forms

	Standard form	Factored form	Vertex form
Equation	$y = a x^{2} + b x + c$	$y = a (x - r) (x - s)$	$y = a (x - h)^{2} + k$
Concavity	Concave up if $a > 0$ Concave down if $a < 0$	Concave up if $a > 0$ Concave down if $a < 0$	Concave up if $a > 0$ Concave down if $a < 0$
Is it easy to find the vertex?	No	No	Yes: $(h, k)$
Is it easy to find the minimum/maximum value?	No	No	Yes: If $a > 0$ , $y = k$ is the minimum value If $a < 0$ , $y = k$ is the maximum value
Is it easy to find the $x$ -intercepts?	No	Yes: $x = r$ or $x = s$	No
Is it easy to find the equation of axis of symmetry?	No	No	Yes: $x = h$

Explore this!

Explore this!

Direction of opening

Vertex

Maximum / minimum value

x-intercept(s)

y-intercept

Axis of symmetry

Notebook

Sketching graphs from factored form

Step 1: Determine the x -intercepts

Step 2: Determine the x value of the vertex (the axis of symmetry value).

Step 3: Substitute the x -value of the vertex into the original equation to find the y -value of the vertex.

Step 4: Sketch if necessary.

Example 1

Example 2

Example 3

Information from equations of a quadratic

Notebook

Graphing with the step method

Notebook

Self-check

Math journal

Portfolio

Summary of diagnosing transformations and quadratic equations

Review of the forms

Step 1: Determine the $x$ -intercepts

Step 2: Determine the $x$ value of the vertex (the axis of symmetry value).

Step 3: Substitute the $x$ -value of the vertex into the original equation to find the $y$ -value of the vertex.