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Algebra 1

7.7.3 The Domain, Vertex, and Zero of Quadratic Functions

Algebra 17.7.3 The Domain, Vertex, and Zero of Quadratic Functions

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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 1 - 4.

The area of a rectangle with a perimeter of 2525 meters and a side length xx: A(x)=x·(252x)2A(x)=x·(252x)2
A parabola on a coordinate grid. The x-axis represents the length in meters of a rectangle and the y-axis represents the its area in square meters. The x-axis scale is 2.5 and extends from 0 to 15. The y-axis scale is 10 and extends from 0 to 40.
1.

What is an inequality that represents the domain?

2.

What is an inequality that represents the range?

3.

What is the vertex of the graph?

4.

What are the zeros of the graph?

Use the following information to answer 5 - 8.

The number of squares as a function of step number nn: f(n)=n2+4f(n)=n2+4
A parabola on a coordinate grid. The x-axis represents the step number of a pattern and the y-axis represents the number of squares in each step. The x-axis scale is 3 and extends from negative 12 to 12. The y-axis scale is 20 and extends from negative 40 to 200.
5.

What is an inequality that represents the domain?

6.

What is an inequality that represents the range?

7.

What is the vertex of the graph?

8.

What are the zeros of the graph?

Use the following information to answer 9 - 12.

The distance in feet that an object has fallen tt seconds after being dropped: g(t)=16t2g(t)=16t2
A parabola on a coordinate grid. The x-axis represents the time in seconds and the y-axis represents the distance fallen 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.
9.

What is an inequality that represents the domain?

10.

What is an inequality that represents the range?

11.

What is the vertex of the graph?

12.

What are the zeros of the graph?

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.

Self Check

Find the approximate domain of the graph below:

Graph of the quadratic function. h(t)=5+60t−16t^2 on a coordinate plane, origin O. Horizontal axis scale 0 to 4 by 1’s, labeled “time (seconds)”. Vertical axis scale 0 to 80 by 20’s, labeled “distance above ground (feet)”. Some of the points of this function are (0 comma 5), (1 comma 49), to a maximum near (1 point 9 comma 61 point 2 5) then decreasing through (2 comma 61), (3 comma 41) and (3.8 comma 0).

  1. x 0
  2. 0 x 3.8
  3. 0 x 60
  4. 0 x 1.8

Additional Resources

The Meaning of Quadratic Characteristics

The height in feet of an object tt seconds after being dropped is graphed below: h(t)=57616t2h(t)=57616t2.

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 tt values, 0t60t6 or only numbers from 00 up through 66. 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 66 seconds after being dropped, so values greater than 66 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)(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 66. It tells us that the height of the object is 00 feet at 66 seconds after being dropped, or the time when the object hits the ground.

Try it

Try It: The Meaning of Quadratic Characteristics

For questions 1 - 2, use the following graph.

Describe the zeros of graph.

Describe what the zeros mean for the graph.

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