Sketching a Graph of a Quadratic Function Using at Least Three Identifiable Points

7.11.3 Sketching a Graph of a Quadratic Function Using at Least Three Identifiable Points

7.11.3 • Sketching a Graph of a Quadratic Function Using at Least Three Identifiable Points

Activity

1. The functions $f$ , $g$ , and $h$ are given. Refer to these functions to complete questions a–f.

Equation	$x$ -intercept	$x$ -coordinate of the vertex
$f (x) = (x + 3) (x - 5)$	a. _____	b. _____
$g (x) = 2 x (x - 3)$	c. _____	d. _____
$h (x) = (x + 4) (4 - x)$	e. _____	f. _____

a. Predict the $x$ -intercepts for function $f (x) = (x + 3) (x - 5)$ .

Compare your answer:

$x$ -intercepts for function $f$ : $(- 3, 0)$ and $(5, 0)$

b. Predict the $x$ -coordinate of the vertex for function $f (x) = (x + 3) (x - 5)$ .

c. Predict the $x$ -intercepts for function $g (x) = 2 x (x - 3)$ .

Compare your answer:

$x$ -intercepts for function $g$ : $(0, 0)$ and $(3, 0)$

d. Predict the $x$ -coordinate of the vertex for function $g (x) = 2 x (x - 3)$ .

1.5

e. Predict the $x$ -intercepts for function $h (x) = (x + 4) (4 - x)$ .

Compare your answer:

$x$ -intercepts for function $h$ : $(- 4, 0)$ and $(4, 0)$

f. Predict the $x$ -coordinate of the vertex for function $h (x) = (x + 4) (4 - x)$ .

2. Use the graphing tool or technology outside the course. Graph the functions $f$ , $g$ , and $h$ :
$f (x) = (x + 3) (x - 5)$
$g (x) = 2 x (x - 3)$
$h (x) = (x + 4) (4 - x)$

Use the graphs to check your predictions.

Access multimedia content

Compare your answer:

3 graphs of the functions f, g, and h are shown on the same coordinate plane. Function f is graphed in blue, function g is given in green, and function h is given in red. The x-axis extends from negative 6 to 6 with a scale of 1. The y-axis has a scale of 10 and extends from negative 20 to 40.

3. Without using technology, sketch a graph that represents the equation $y = (x - 7) (x + 11)$ and that shows the $x$ -intercepts and the vertex. Think about how to find the $y$ -coordinate of the vertex. Be prepared to explain your reasoning.

Compare your answer:

The $x$ -intercepts for $(x - 7) (x + 11)$ are at $x = 7$ and $x = - 11$ . The vertex for $(x - 7) (x + 11)$ is halfway between the $x$ -intercepts, at $x = - 2$ . If the $x$ -coordinate of the vertex is -2, then the $y$ -value is $(- 2, - 7) (- 2 + 11)$ or -81. The vertex is at $(- 2, 81)$ . The $y$ -value when $x = 0$ is $(0 - 7) (0 + 11)$ , which is -77. The $y$ -intercept is $(0, - 77)$ .

A parabola on a coordinate grid. The x-axis scale is 2 and extends from negative 12 to 12. The y-axis scale is 20 and extends from negative 100 to 100.

Are you ready for more?

Extending Your Thinking

The quadratic function $f$ is given by $f (x) = x^{2} + 2 x + 6$ .

Find $f (- 2)$ .

Compare your answer:

Find $f (0)$ .

Compare your answer:

What is the $x$ -coordinate of the vertex of the graph of this quadratic function?

Compare your answer:

-1

Does the graph have any $x$ -intercepts? Explain or show how you know.

Compare your answer:

No, the graph does not have any $x$ -intercepts. The $y$ -coordinate of the vertex is 5 because $f (- 1) = (- 1) 2 + 2 (- 1) + 6 = 5$ . The coefficient of the squared term is positive, so the vertex is a minimum, which means the output of the function is never less than 5.

Video: Sketching a Graph of a Quadratic Function

Watch the following video to learn more about sketching a graph of a quadratic function using at least three identifiable points.

Access multimedia content

Self Check

Find the coordinate of the vertex of

f(x) = (x - 2)(x - 4)

$(0, 8)$
$(4, 0)$
$(3, -1)$
$(2, 0)$

Additional Resources

Graphing Quadratics With Points

The function $f$ given by $f (x) = (x + 1) (x - 1)$ is written in factored form. Recall that this form is helpful for finding the zeros of the function (where the function has the value 0) and telling us the $x$ -intercepts on the graph representing the function.

Here is a graph representing $f$ . It shows two $x$ -intercepts at $x = - 1$ and $x = 3$ .

If we use –1 and 3 as inputs to $f$ , what are the outputs?

$f (- 1) = (- 1 + 1) (- 1 - 3) = (0) (- 4) = 0$
$f (3) = (3 + 1) (3 - 3) = (4) (0) = 0$

Because the inputs –1 and 3 produce an output of 0, they are the zeros of the function $f$ . And because both $x$ values have 0 for their $y$ value, they also give us the $x$ -intercepts of the graph (the points where the graph crosses the $x$ -axis, which always have a $y$ -coordinate of 0). So, the zeros of a function have the same values as the $x$ -coordinates of the $x$ -intercepts of the graph of the function.

The factored form can also help us identify the vertex of the graph, which is the point where the function reaches its minimum value. Notice that the $x$ -coordinate of the vertex is 1, and that 1 is halfway between –1 and 3. Once we know the $x$ -coordinate of the vertex, we can find the $y$ -coordinate by evaluating the function $f (1) = (1 + 1) (1 - 3) = (2) (- 2) = - 4$ . So the vertex is at $(1, - 4)$ .

When a quadratic function is in standard form, the $y$ -intercept is clear: its $y$ -coordinate is the constant term $c$ in $a x^{2} + b x + c$ . To find the $y$ -intercept from factored form, we can evaluate the function at $x = 0$ , because the $y$ -intercept is the point where the graph has an input value of 0:

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

Try it

Graphing Quadratics with Points

Find the zeros and vertex of the function $f$ given by $f (x) = (x - 4) (x + 2)$ . Use the points to graph $f$ .

Here is how to find the function’s zeros and vertex:

Step 1 - Set each factor equal to 0 and solve.

$x - 4 = 0, x = 4$
$x + 2 = 0, x = - 2$

Step 2 - Write the zeros as points. $(4, 0) (- 2, 0)$

Step 3 - Find the $x$ -value halfway between the zeros. $x = 1$

Step 4 - Substitute $x = 1$ into the function to find the vertex. $f (1) = (1 - 4) (1 + 2) = (- 3) (3) = - 9$ Vertex: $(1, - 9)$

Step 5 - Now, use these points to graph $f$ :

A graph of a parabola opening upwards with its vertex at (1, negative 9) and zeros at (4, 0) and (negative 2, 0).