The quadratic function shown graphed below has the form f (x)=(x+h)´ +k. Determine the values of h and k. Point P is the larger zero of f(x). Algebraically determine its value to the nearest hundredth. Show how you arrived at your answer.
The quadratic function shown graphed below has the form f (x)=(x+h)´ +k. Determine the values of h and k. Point P is the larger zero of f(x). Algebraically determine its value to the nearest hundredth. Show how you arrived at your answer.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![### Quadratic Function Analysis
#### Problem Statement
The quadratic function shown graphed below has the form \( f(x) = (x + h)^2 + k \).
- Determine the values of \( h \) and \( k \).
- Point \( P \) is the larger zero of \( f(x) \). Algebraically determine its value to the nearest hundredth. Show how you arrived at your answer.
#### Graph Description
The graph depicts a quadratic function in standard form. It is a parabola that opens upwards. The graph intersects the x-axis at two points.
### Steps to Solve:
##### 1. Identify the Vertex
The vertex form of a quadratic function is given by \( f(x) = (x + h)^2 + k \).
From the graph:
- The vertex (minimum point of the parabola) is at \( (-4, -5) \).
Thus, the vertex gives us the values of \( h \) and \( k \):
- \( h = -4 \) (Note: the value h represents a horizontal shift, \( x + h \) becomes \( x + 4 \) when \( h = -4 \))
- \( k = -5 \).
Therefore, the function is:
\[ f(x) = (x + 4)^2 - 5 \]
##### 2. Find the Zeros of the Function
To find the zeros, set \( f(x) = 0 \).
\[ 0 = (x + 4)^2 - 5 \]
Solve for \( x \):
\[ (x + 4)^2 = 5 \]
\[ x + 4 = \pm \sqrt{5} \]
\[ x = -4 + \sqrt{5} \quad \text{or} \quad x = -4 - \sqrt{5} \]
Calculate the values to the nearest hundredth:
\[ x = -4 + \sqrt{5} \approx -4 + 2.24 = -1.76 \]
\[ x = -4 - \sqrt{5} \approx -4 - 2.24 = -6.24 \]
##### 3. Identify Point \( P \)
Point \( P \) is the larger zero:
\[ P \approx -1.76 \]
### Conclusion
- The vertex values are \( h = -4 \), \( k = -5 \).
- The larger zero](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0f40088-de79-422a-8e62-fd0321faa220%2Fd7e32359-3065-4930-8d54-979fb395b97b%2Fh87nmkf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Quadratic Function Analysis
#### Problem Statement
The quadratic function shown graphed below has the form \( f(x) = (x + h)^2 + k \).
- Determine the values of \( h \) and \( k \).
- Point \( P \) is the larger zero of \( f(x) \). Algebraically determine its value to the nearest hundredth. Show how you arrived at your answer.
#### Graph Description
The graph depicts a quadratic function in standard form. It is a parabola that opens upwards. The graph intersects the x-axis at two points.
### Steps to Solve:
##### 1. Identify the Vertex
The vertex form of a quadratic function is given by \( f(x) = (x + h)^2 + k \).
From the graph:
- The vertex (minimum point of the parabola) is at \( (-4, -5) \).
Thus, the vertex gives us the values of \( h \) and \( k \):
- \( h = -4 \) (Note: the value h represents a horizontal shift, \( x + h \) becomes \( x + 4 \) when \( h = -4 \))
- \( k = -5 \).
Therefore, the function is:
\[ f(x) = (x + 4)^2 - 5 \]
##### 2. Find the Zeros of the Function
To find the zeros, set \( f(x) = 0 \).
\[ 0 = (x + 4)^2 - 5 \]
Solve for \( x \):
\[ (x + 4)^2 = 5 \]
\[ x + 4 = \pm \sqrt{5} \]
\[ x = -4 + \sqrt{5} \quad \text{or} \quad x = -4 - \sqrt{5} \]
Calculate the values to the nearest hundredth:
\[ x = -4 + \sqrt{5} \approx -4 + 2.24 = -1.76 \]
\[ x = -4 - \sqrt{5} \approx -4 - 2.24 = -6.24 \]
##### 3. Identify Point \( P \)
Point \( P \) is the larger zero:
\[ P \approx -1.76 \]
### Conclusion
- The vertex values are \( h = -4 \), \( k = -5 \).
- The larger zero
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