10. Write the equation of the quadratic graphed below 10 10 -10 (0,5) (2,3) 0 -5 -10- ((116) 5
10. Write the equation of the quadratic graphed below 10 10 -10 (0,5) (2,3) 0 -5 -10- ((116) 5
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 41E
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![### Quadratic Equations: Graph Analysis
**Question:**
10. Write the equation of the quadratic graphed below.
**Graph Description:**
The graph presented is of a quadratic equation that forms a parabola opening downwards. It depicts several key points:
- The vertex of the parabola is at \((-2, 9)\).
- The y-intercept is \((0, 5)\).
- Another point shown on the parabola is \((2, 5)\).
**Steps to Determine the Equation:**
1. **Identify the Vertex Form of a Quadratic Equation:**
The vertex form of a quadratic equation is given by:
\[
y = a(x-h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola.
2. **Substitute the Vertex Coordinates:**
For the given vertex \((-2, 9)\):
\[
y = a(x + 2)^2 + 9
\]
3. **Use Another Point to Find 'a':**
Substitute the point \((0, 5)\) into the equation to solve for \(a\):
\[
5 = a(0 + 2)^2 + 9
\]
This simplifies to:
\[
5 = 4a + 9
\]
Solving for \(a\):
\[
4a = 5 - 9
\]
\[
4a = -4
\]
\[
a = -1
\]
4. **Write the Final Equation:**
Substitute \(a = -1\) into the vertex form equation:
\[
y = - (x + 2)^2 + 9
\]
Hence, the equation of the quadratic graph is:
\[
y = - (x + 2)^2 + 9
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc1123ec-a0e0-4e35-afe0-df78f47a58b5%2Feeff3162-3894-4f22-a326-f092465ee9bc%2F7h59t1c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Quadratic Equations: Graph Analysis
**Question:**
10. Write the equation of the quadratic graphed below.
**Graph Description:**
The graph presented is of a quadratic equation that forms a parabola opening downwards. It depicts several key points:
- The vertex of the parabola is at \((-2, 9)\).
- The y-intercept is \((0, 5)\).
- Another point shown on the parabola is \((2, 5)\).
**Steps to Determine the Equation:**
1. **Identify the Vertex Form of a Quadratic Equation:**
The vertex form of a quadratic equation is given by:
\[
y = a(x-h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola.
2. **Substitute the Vertex Coordinates:**
For the given vertex \((-2, 9)\):
\[
y = a(x + 2)^2 + 9
\]
3. **Use Another Point to Find 'a':**
Substitute the point \((0, 5)\) into the equation to solve for \(a\):
\[
5 = a(0 + 2)^2 + 9
\]
This simplifies to:
\[
5 = 4a + 9
\]
Solving for \(a\):
\[
4a = 5 - 9
\]
\[
4a = -4
\]
\[
a = -1
\]
4. **Write the Final Equation:**
Substitute \(a = -1\) into the vertex form equation:
\[
y = - (x + 2)^2 + 9
\]
Hence, the equation of the quadratic graph is:
\[
y = - (x + 2)^2 + 9
\]
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