The Domain, Vertex, and Zero of Quadratic Functions

7.7.3 The Domain, Vertex, and Zero of Quadratic Functions

7.7.3 • The Domain, Vertex, and Zero of Quadratic Functions

Activity

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

Here are three sets of descriptions and equations that represent some familiar quadratic functions. The graphs show what a graphing technology may produce when the equations are graphed. For a more accurate answer, use the graphing tool or technology outside the course. Graph the equation that represents this scenario using the Desmos tool below. (Students had access to Desmos.)

Describe a domain that is appropriate for the situation. Think about any upper or lower limits for the input, as well as whether all numbers make sense as the input. Write the domain as an inequality. Then, describe how the graph should be modified to show the domain that makes sense.
Describe a range that is appropriate for the situation. Think about any upper or lower limits for the input, as well as whether all numbers make sense as the input. Write the range as an inequality. What does the range mean in this situation? Then, describe how the graph should be modified to show the range that makes sense.
Identify or estimate the The vertex (of a graph). Describe what it means in the situation.
Identify or estimate the zero of the function. Describe what they mean in the situation.

Use the following information to answer questions 1 – 4.

The area of a rectangle with a perimeter of $25$ meters and a side length $x$ : $A (x) = x \cdot \frac{(25 - 2 x)}{2}$

What is an inequality that represents the domain?

Compare your answer:

$0 < x < 12.5$ . All positive rational numbers make sense as an input. The part of the graph that is below the horizontal line is not meaningful here (negative lengths or lengths greater than $12.5$ produce a negative area, which doesn’t make sense). Having a length or area equal to $0$ doesn’t make sense either.

What is an inequality that represents the range?

Compare your answer:

$0 < y < 39.06$ . All positive rational numbers make sense as an output up to the maximum of the graph at 39.06. The part of the graph that is below the horizontal line is not meaningful here since there cannot be a negative area (which doesn’t make sense). Having a length or area equal to 0 doesn’t make sense either. The range gives the minimum and maximum values possible for the area from a perimeter of 25.

What is the vertex of the graph?

Compare your answer:

The vertex is the point on the graph when the input is at $6.25$ . It’s around $(6.25, 39.063)$ . It’s when the rectangle has the greatest area.

What are the zeros of the graph?

Compare your answer:

The zeros are $0$ and $12.5$ . They tell us that when the side length of the rectangle is $0$ or $12.5$ meters, the rectangle has no area (no rectangle exists).

Use the following information to answer questions 5 – 8.

The number of squares as a function of step number $n$ : $f (n) = n^{2} + 4$

What is an inequality that represents the domain?

Compare your answer:

$n \geq 0$ where $n$ is a whole number. Step numbers that are negative or decimal don’t make sense here. The graph should show points at non-negative integer values (instead of a continuous curve). Everything to the left of the vertical axis does not have meaning here.

What is an inequality that represents the range?

Compare your answer:

$f \geq 4$ . The number of squares starts with 4 and increases.

What is the vertex of the graph?

Compare your answer:

The vertex is $(0, 4)$ . It tells us the number of squares at Step $0$ .

What are the zeros of the graph?

Compare your answer:

There are no zeros for this function. There is not a time when there are no squares in the pattern.

Use the following information to answer questions 9 - 12.

The distance in feet that an object has fallen $t$ seconds after being dropped: $g (t) = 16 t^{2}$

What is an inequality that represents the domain?

Compare your answer:

$t \geq 0$ or only positive numbers and $0$ . Negative values of time don’t make sense here, so the part of the graph to the left side of the vertical axis has no meaning.

10.

What is an inequality that represents the range?

Compare your answer:

$g \geq 0$ . The distance fallen increases as time increases and would start at $g = 0$ .

11.

What is the vertex of the graph?

Compare your answer:

The vertex is $(0, 0)$ . It tells us the distance fallen at $0$ seconds, which is $0$ feet.

12.

What are the zeros of the graph?

Compare your answer:

The vertex also tells us the zero of the function. It’s the only time when the distance fallen is $0$ feet.

Video: Identifying Domain, Vertex, and Zero of Quadratic Functions

Watch the following video to learn more about the domain, vertex, and zero of quadratic functions.

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Self Check

Find the approximate domain of the graph below:

$x \geq 0$
$0 \leq x \leq 3.8$
$0 \leq x \leq 60$
$0 \leq x \leq 1.8$

Additional Resources

The Meaning of Quadratic Characteristics

The height in feet of an object $t$ seconds after being dropped is graphed below: $h (t) = 576 - 16 t^{2}$ .

A parabola on a coordinate grid. The x-axis represents the time in seconds and the y-axis represents the height in feet. The x-axis scale is 2 and extends from negative 10 to 10. The y-axis scale is 100 and extends from negative 200 to 1,000.

Find the domain of the function.

The domain is the $t$ values, $0 \leq t \leq 6$ or only numbers from $0$ up through $6$ . Negative values of time don’t make sense here, so the part of the graph to the left side of the vertical axis has no meaning. The object hits the ground $6$ seconds after being dropped, so values greater than $6$ are not meaningful.

Find the range of the function.

The range of the height of the function is from the initial height down to the ground. In this case, since the object is being dropped from a height of 576 feet and falls to the ground, the range of the function is [0, 576]. This means that the object's height will be between 0 feet (when it hits the ground) and 576 feet (when it is initially dropped).

Find the vertex and describe its meaning.

The vertex (of a graph) is the peak of the parabola, or the maximum, at the point $(0, 576)$ . It tells us the time when the height is the greatest.

Find the zeros and describe their meaning.

The zero of the function is $6$ . It tells us that the height of the object is $0$ feet at $6$ seconds after being dropped, or the time when the object hits the ground.

Try it

The Meaning of Quadratic Characteristics

For questions 1 – 2, use the following graph.

A graph illustrates height (feet) over time (seconds) with a parabolic curve. The object starts at 0 feet at 0 seconds, reaches a maximum height of approximately 64 feet at 8 seconds, and returns to 0 feet at about 17 seconds.

Describe the zeros of graph.

Compare your answer: Your answer may vary, but here is a sample.

$x = 0$ , $x = 16$

Describe what the zeros mean for the graph.

Compare your answer: Your answer may vary, but here is a sample.

Looking at the label on the $x$ -axis, this value means that at 0 seconds and 16 seconds, the height is 0 feet.