Welcome to Westonci.ca, your one-stop destination for finding answers to all your questions. Join our expert community now! Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To solve for the vertex and x-intercepts for the quadratic function [tex]\( f(x) = -3x^2 - 7x + 2 \)[/tex], we'll follow these mathematical steps.
### Step 1: Find the Vertex
The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found using the vertex formula:
- Vertex x-coordinate ([tex]\( x_v \)[/tex]): [tex]\( x_v = -\frac{b}{2a} \)[/tex]
Given:
- [tex]\( a = -3 \)[/tex]
- [tex]\( b = -7 \)[/tex]
Substituting the values into the formula:
[tex]\[ x_v = -\frac{-7}{2(-3)} = \frac{7}{6} \approx -1.17 \][/tex]
Next, to find the y-coordinate of the vertex ([tex]\( y_v \)[/tex]), substitute [tex]\( x_v \)[/tex] back into the original function:
[tex]\[ y_v = f(x_v) = -3(x_v)^2 - 7(x_v) + 2 \][/tex]
Substituting [tex]\( x_v = -1.17 \)[/tex]:
[tex]\[ y_v = -3(-1.17)^2 - 7(-1.17) + 2 \][/tex]
[tex]\[ y_v \approx 6.08 \][/tex]
So, the vertex of the quadratic function is approximately:
[tex]\[ \text{Vertex: } (-1.17, 6.08) \][/tex]
### Step 2: Find the x-intercepts
To find the x-intercepts (where the function crosses the x-axis), set [tex]\( f(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ -3x^2 - 7x + 2 = 0 \][/tex]
To solve this quadratic equation, we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here:
- [tex]\( a = -3 \)[/tex]
- [tex]\( b = -7 \)[/tex]
- [tex]\( c = 2 \)[/tex]
First, calculate the discriminant ([tex]\( \Delta \)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-7)^2 - 4(-3)(2) \][/tex]
[tex]\[ \Delta = 49 + 24 \][/tex]
[tex]\[ \Delta = 73 \][/tex]
Since the discriminant is positive, there are two real x-intercepts. Now, substitute the values back into the quadratic formula:
[tex]\[ x_1 = \frac{-(-7) + \sqrt{73}}{2(-3)} \][/tex]
[tex]\[ x_1 = \frac{7 + \sqrt{73}}{-6} \][/tex]
[tex]\[ x_1 \approx -2.59 \][/tex]
[tex]\[ x_2 = \frac{-(-7) - \sqrt{73}}{2(-3)} \][/tex]
[tex]\[ x_2 = \frac{7 - \sqrt{73}}{-6} \][/tex]
[tex]\[ x_2 \approx 0.26 \][/tex]
So, the x-intercepts of the quadratic function are approximately:
[tex]\[ x_1 \approx -2.59 \][/tex]
[tex]\[ x_2 \approx 0.26 \][/tex]
### Final answer:
- Vertex: [tex]\((-1.17, 6.08)\)[/tex]
- x-intercepts: [tex]\([-2.59, 0.26]\)[/tex]
### Step 1: Find the Vertex
The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found using the vertex formula:
- Vertex x-coordinate ([tex]\( x_v \)[/tex]): [tex]\( x_v = -\frac{b}{2a} \)[/tex]
Given:
- [tex]\( a = -3 \)[/tex]
- [tex]\( b = -7 \)[/tex]
Substituting the values into the formula:
[tex]\[ x_v = -\frac{-7}{2(-3)} = \frac{7}{6} \approx -1.17 \][/tex]
Next, to find the y-coordinate of the vertex ([tex]\( y_v \)[/tex]), substitute [tex]\( x_v \)[/tex] back into the original function:
[tex]\[ y_v = f(x_v) = -3(x_v)^2 - 7(x_v) + 2 \][/tex]
Substituting [tex]\( x_v = -1.17 \)[/tex]:
[tex]\[ y_v = -3(-1.17)^2 - 7(-1.17) + 2 \][/tex]
[tex]\[ y_v \approx 6.08 \][/tex]
So, the vertex of the quadratic function is approximately:
[tex]\[ \text{Vertex: } (-1.17, 6.08) \][/tex]
### Step 2: Find the x-intercepts
To find the x-intercepts (where the function crosses the x-axis), set [tex]\( f(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ -3x^2 - 7x + 2 = 0 \][/tex]
To solve this quadratic equation, we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here:
- [tex]\( a = -3 \)[/tex]
- [tex]\( b = -7 \)[/tex]
- [tex]\( c = 2 \)[/tex]
First, calculate the discriminant ([tex]\( \Delta \)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-7)^2 - 4(-3)(2) \][/tex]
[tex]\[ \Delta = 49 + 24 \][/tex]
[tex]\[ \Delta = 73 \][/tex]
Since the discriminant is positive, there are two real x-intercepts. Now, substitute the values back into the quadratic formula:
[tex]\[ x_1 = \frac{-(-7) + \sqrt{73}}{2(-3)} \][/tex]
[tex]\[ x_1 = \frac{7 + \sqrt{73}}{-6} \][/tex]
[tex]\[ x_1 \approx -2.59 \][/tex]
[tex]\[ x_2 = \frac{-(-7) - \sqrt{73}}{2(-3)} \][/tex]
[tex]\[ x_2 = \frac{7 - \sqrt{73}}{-6} \][/tex]
[tex]\[ x_2 \approx 0.26 \][/tex]
So, the x-intercepts of the quadratic function are approximately:
[tex]\[ x_1 \approx -2.59 \][/tex]
[tex]\[ x_2 \approx 0.26 \][/tex]
### Final answer:
- Vertex: [tex]\((-1.17, 6.08)\)[/tex]
- x-intercepts: [tex]\([-2.59, 0.26]\)[/tex]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
