Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Quadratic Function Analysis
#### Given Equation:
\( y = -2(x+1)(x+3) \)
#### Quadratic Form:
The equation is in the factored form.
\[ y = a(x-r_1)(x-r_2) \]
where \( a = -2 \), \( r_1 = -1 \), and \( r_2 = -3 \).
#### Vertex:
To find the vertex, use the formula to find the x-coordinate:
\[ x = \frac{r_1 + r_2}{2} \]
\[ x = \frac{-1 + (-3)}{2} \]
\[ x = -2 \]
Substituting \( x = -2 \) into the equation to find \( y \):
\[ y = -2(-2+1)(-2+3) \]
\[ y = -2(-1)(1) \]
\[ y = 2 \]
So, the vertex is at \((-2, 2)\).
#### x-intercepts:
The x-intercepts (roots) are the values of \( x \) when \( y = 0 \):
\[ (x+1)(x+3) = 0 \]
So the x-intercepts are \( x = -1 \) and \( x = -3 \).
#### y-intercept:
The y-intercept is the value of \( y \) when \( x = 0 \):
\[ y = -2(0+1)(0+3) \]
\[ y = -2(1)(3) \]
\[ y = -6 \]
So, the y-intercept is at \( y = -6 \).
#### Axis of Symmetry:
The axis of symmetry is the vertical line that passes through the vertex. It can be expressed as:
\[ x = -2 \]
#### Domain:
The domain of the quadratic function is all real numbers:
\[ (-\infty, +\infty) \]
#### Range:
Since the parabola opens downwards (negative coefficient), the range is:
\[ (-\infty, 2] \]
### Graph Explanation:
The graph is a coordinate plane with an x-axis and a y-axis. The x-axis ranges from -10 to 10, and the y-axis ranges from -10 to 10, marked at single unit intervals. The vertex of the parabola should be plotted at \((-2,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5d4eb86-ab39-40ab-82d8-e64b0c4f1834%2F4c2d5437-e0cc-4322-85ea-1ad81eba3096%2Fgtx9i6kn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Quadratic Function Analysis
#### Given Equation:
\( y = -2(x+1)(x+3) \)
#### Quadratic Form:
The equation is in the factored form.
\[ y = a(x-r_1)(x-r_2) \]
where \( a = -2 \), \( r_1 = -1 \), and \( r_2 = -3 \).
#### Vertex:
To find the vertex, use the formula to find the x-coordinate:
\[ x = \frac{r_1 + r_2}{2} \]
\[ x = \frac{-1 + (-3)}{2} \]
\[ x = -2 \]
Substituting \( x = -2 \) into the equation to find \( y \):
\[ y = -2(-2+1)(-2+3) \]
\[ y = -2(-1)(1) \]
\[ y = 2 \]
So, the vertex is at \((-2, 2)\).
#### x-intercepts:
The x-intercepts (roots) are the values of \( x \) when \( y = 0 \):
\[ (x+1)(x+3) = 0 \]
So the x-intercepts are \( x = -1 \) and \( x = -3 \).
#### y-intercept:
The y-intercept is the value of \( y \) when \( x = 0 \):
\[ y = -2(0+1)(0+3) \]
\[ y = -2(1)(3) \]
\[ y = -6 \]
So, the y-intercept is at \( y = -6 \).
#### Axis of Symmetry:
The axis of symmetry is the vertical line that passes through the vertex. It can be expressed as:
\[ x = -2 \]
#### Domain:
The domain of the quadratic function is all real numbers:
\[ (-\infty, +\infty) \]
#### Range:
Since the parabola opens downwards (negative coefficient), the range is:
\[ (-\infty, 2] \]
### Graph Explanation:
The graph is a coordinate plane with an x-axis and a y-axis. The x-axis ranges from -10 to 10, and the y-axis ranges from -10 to 10, marked at single unit intervals. The vertex of the parabola should be plotted at \((-2,
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