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]
