## Modeling With Quadratic Functions 24 Exercises

**Vocabulary and Core Concept Check**

Question 1.**WRITING** Explain when it is appropriate to use a quadratic model for a set of data.Answer:

x-intercepts of 12 and -6 passes through Answer:

x-intercepts of 9 and 1 passes through Answer:

x-intercepts of -16 and -2 passes through Answer:

x-intercepts of -7 and -3 passes through Answer:

Question 15.**WRITING** Explain when to use intercept form and when to use vertex form when writing an equation of a parabola.Answer:

**ANALYZING EQUATIONS** Which of the following equations represent the parabola?A. y = 2B. y = 22 4.5C. y = 22 4.5D. y = 2Answer:

**In Exercises 1720, write an equation of the parabola in vertex form or intercept form.**

Question 17.

**ERROR ANALYSIS** Describe and correct the error in writing an equation of the parabola.Answer:

Question 22.**MATHEMATICAL CONNECTIONS** The area of a rectangle is modeled by the graph where y is the area and x is the width . Write an equation of the parabola. Find the dimensions and corresponding area of one possible rectangle. What dimensions result in the maximum area?Answer:

Question 23.**MODELING WITH MATHEMATICS** Every rope has a safe working load. A rope should not be used to lift a weight greater than its safe working load. The table shows the safe working loads S for ropes with circumference C . Write an equation for the safe working load for a rope. Find the safe working load for a rope that has a circumference of 10 inches.Answer:

Question 29.

Answer:

**Mathematical Practices**

**Performance Task**

## Focus Of A Parabola 23 Exercises

**Vocabulary and Core Concept Check**

Question 1.**COMPLETE THE SENTENCE** A parabola is the set of all points in a plane equidistant from a fixed point called the ______ and a fixed line called the __________ .Answer:

Question 2.**WRITING **Explain how to find the coordinates of the focus of a parabola with vertex and directrix y = 5.Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 310, use the Distance Formula to write an equation of the parabola.**

Question 3.

Answer:

Question 47.**MODELING WITH MATHEMATICS** Scientists studying dolphin echolocation simulate the projection of a bottlenose dolphins clicking sounds using computer models. The models originate the sounds at the focus of a parabolic reflector. The parabola in the graph shows the cross section of the reflector with focal length of 1.3 inches and aperture width of 8 inches. Write an equation to represent the cross section of the reflector. What is the depth of the reflector?Answer:

Question 48.**MODELING WITH MATHEMATICS** Solar energy can be concentrated using long troughs that have a parabolic cross section as shown in the figure. Write an equation to represent the cross section of the trough. What are the domain and range in this situation? What do they represent?Answer:

Question 49.**ABSTRACT REASONING** As | p | increases, how does the width of the graph of the equation y = \x2 change? Explain your reasoning.Answer:

**Write an equation of the line that passes through the points.**

Question 55.

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**Describe the transformation of f = x2 represented by g. Then graph each function.**

Question 1.

**Write an equation for the parabola with the given characteristics.**

Question 13.passes through and has vertex

Question 14.passes through and has x-intercepts of -1 and 5

Question 15.passes through , , and

Question 16.The table shows the heights y of a dropped object after x seconds. Verify that the data show a quadratic relationship. Write a function that models the data. How long is the object in the air?

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## Lesson 23 Focus Of A Parabola

**Essential QuestionWhat is the focus of a parabola?EXPLORATION 1Analyzing Satellite DishesWork with a partner**. Vertical rays enter a satellite dish whose cross section is a parabola. When the rays hit the parabola, they reflect at the same angle at which they entered. a. Draw the reflected rays so that they intersect the y-axis.b. What do the reflected rays have in common?c. The optimal location for the receiver of the satellite dish is at a point called the focus of the parabola. Determine the location of the focus. Explain why this makes sense in this situation.

**EXPLORATION 2Analyzing Spotlights****Work with a partner.** Beams of light are coming from the bulb in a spotlight, located at the focus of the parabola. When the beams hit the parabola, they reflect at the same angle at which they hit. Draw the reflected beams. What do they have in common? Would you consider this to be the optimal result? Explain.

**Communicate Your Answer**

What is the focus of a parabola?

Question 4.Describe some of the properties of the focus of a parabola.

**2.3 Lesson**

**Monitoring Progress**

Question 1.Use the Distance Formula to write an equation of the parabola with focus F and directrix y = 3.

**Identify the focus, directrix, and axis of symmetry of the parabola. Then graph the equation.**

Question 2.

focus: )

**Monitoring Progress**

Write an equation of a parabola with vertex and focus .

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