Explanation Result used: If the general quadratic equation is a x 2 + b x + c , then the x intercept is x = − b 2 a . Calculation: The given equation f ( x ) = − 2 3 ( x + 1 2 ) 2 + 10 can be written as follows, f ( x ) = − 2 3 ( x 2 + 2 ⋅ 1 2 x + 1 4 ) + 10 = ( − 2 3 ⋅ x 2 ) + ( − 2 3 ⋅ x ) + ( − 2 3 ⋅ 1 4 ) + 10 = − 2 3 x 2 − 2 3 x − 2 12 + 10 On further simplification, f ( x ) = − 2 3 x 2 − 2 3 x − 2 12 + 120 12 = − 2 3 x 2 − 2 3 x + 118 12 = − 2 3 x 2 − 2 3 x + 59 6 Compare the given equation f ( x ) = − 2 3 x 2 − 2 3 x + 59 6 into the standard quadratic equation then a = 1 2 , b = 3 and c = 29 2 . Substitute a = − 2 3 and b = − 2 3 in x = − b 2 a and obtain the x coordinate of the vertex. x = − ( − 2 3 ) 2 ( − 2 3 ) = − 1 2 That is, the x intercept is x = − 1 2 . Substitute x = − 1 2 in f ( x ) = − 2 3 ( x + 1 2 ) 2 + 10 to compute the y coordinate of the vertex. f ( − 1 2 ) = − 2 3 ( − 1 2 + 1 2 ) 2 + 10 = − 2 3 ( 0 ) 2 + 10 = 0 + 10 = 10 That is, the vertex of the parabola is ( − 1 2 , 10 ) . Obtain the x and y intercept of the parabola. Substitute x = 0 in f ( x ) = − 2 3 x 2 − 2 3 x + 59 6 to compute the y intercept. f ( 0 ) = − 2 3 ( 0 ) 2 − 2 3 ( 0 ) + 59 6 = 0 − 0 + 59 6 = 59 6 Therefore, the y intercept is ( 0 , 59 6 ) . Substitute f ( x ) = 0 in f ( x ) = − 2 3 ( x + 1 2 ) 2 + 10 to compute the x intercept. 0 = − 2 3 ( x + 1 2 ) 2 + 10 − 10 = − 2 3 ( x + 1 2 ) 2 − 30 − 2 = ( x + 1 2 ) 2 15 = ( x + 1 2 ) 2 Simplify further as shown below, x + 1 2 = ± 15 x = − 0

4th Edition

ISBN: 9781266240829

Author: sobecki

Publisher: MCG CUSTOM

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 6.5, Problem 23E

To determine

To sketch: The parabola f(x)=−23(x+12)2+10 .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.5, Problem 22E

Chapter 6.5, Problem 24E

Students have asked these similar questions

Now consider equations of the form ×-a=v = √bx + c, where a, b, and c are all positive integers and b>1. (f) Create an equation of this form that has 7 as a solution and an extraneous solution. Give the extraneous solution. (g) What must be true about the value of bx + c to ensure that there is a real number solution to the equation? Explain.

The equation ×+ 2 = √3x+10 is of the form ×+ a = √bx + c, where a, b, and c are all positive integers and b > 1. Using this equation as a model, create your own equation that has extraneous solutions. (d) Using trial and error with numbers for a, b, and c, create an equation of the form x + a = √bx + c, where a, b, and c are all positive integers and b>1 such that 7 is a solution and there is an extraneous solution. (Hint: Substitute 7 for x, and choose a value for a. Then square both sides so you can choose a, b, and c that will make the equation true.) (e) Solve the equation you created in Part 2a.

A basketball player made 12 out of 15 free throws she attempted. She wants to know how many consecutive free throws she would have to make to raise the percent of successful free throws to 85%. (a) Write an equation to represent this situation. (b) Solve the equation. How many consecutive free throws would she have to make to raise her percent to 85%?

Chapter 6 Solutions

MATH IN OUR WORLD (LL) W/18 WEEKS ALEKS

Ch. 6.1 - Translate each verbal statement into symbols. (a)...Ch. 6.1 - Prob. 2TTO Ch. 6.1 - Prob. 3TTO Ch. 6.1 - Prob. 4TTO Ch. 6.1 - Prob. 5TTO Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Whats the difference between checking your answer...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The parabola f ( x ) = − 2 3 ( x + 1 2 ) 2 + 10 .

Solution Summary: The author illustrates the graph of the parabola f(x)=-23.

Concept explainers

Videos

To sketch: The parabola f(x)=−23(x+12)2+10 .

Want to see the full answer?

Chapter 6 Solutions