Certainly! Let's solve this step-by-step.
Given the function:
[tex]\[ f(x) = 2x^2 + 6x - 5 \][/tex]
### a. Finding the Vertex
The vertex of a parabola given by the equation [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula for the x-coordinate of the vertex, which is:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = 2 \)[/tex], [tex]\( b = 6 \)[/tex], and [tex]\( c = -5 \)[/tex].
First, we calculate the x-coordinate of the vertex:
[tex]\[ x = -\frac{6}{2 \cdot 2} = -\frac{6}{4} = -1.5 \][/tex]
Next, we substitute [tex]\( x = -1.5 \)[/tex] back into the function to find the y-coordinate of the vertex:
[tex]\[ f(-1.5) = 2(-1.5)^2 + 6(-1.5) - 5 \][/tex]
[tex]\[ f(-1.5) = 2(2.25) - 9 - 5 \][/tex]
[tex]\[ f(-1.5) = 4.5 - 9 - 5 \][/tex]
[tex]\[ f(-1.5) = -9.5 \][/tex]
Therefore, the vertex is:
[tex]\[ (-1.5, -9.5) \][/tex]
### b. Finding the Vertical Intercept
The vertical intercept (y-intercept) occurs where [tex]\( x = 0 \)[/tex].
Substituting [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = 2(0)^2 + 6(0) - 5 \][/tex]
[tex]\[ f(0) = -5 \][/tex]
Therefore, the vertical intercept is:
[tex]\[ (0, -5) \][/tex]
### c. Finding the x-Intercepts
The x-intercepts occur where [tex]\( y = 0 \)[/tex], i.e., where [tex]\( f(x) = 0 \)[/tex].
For the quadratic function [tex]\( f(x) = 2x^2 + 6x - 5 \)[/tex], we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 2 \)[/tex], [tex]\( b = 6 \)[/tex], and [tex]\( c = -5 \)[/tex].
First, we calculate the discriminant:
[tex]\[ b^2 - 4ac = 6^2 - 4(2)(-5) \][/tex]
[tex]\[ b^2 - 4ac = 36 + 40 \][/tex]
[tex]\[ b^2 - 4ac = 76 \][/tex]
Next, we find the two solutions for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-6 \pm \sqrt{76}}{4} \][/tex]
Which gives us:
[tex]\[ x_1 = \frac{-6 + \sqrt{76}}{4} \approx 0.68 \][/tex]
[tex]\[ x_2 = \frac{-6 - \sqrt{76}}{4} \approx -3.68 \][/tex]
Therefore, the x-intercepts are:
[tex]\[ (0.68, 0) \][/tex]
[tex]\[ (-3.68, 0) \][/tex]
### Final Answers:
- The vertex: [tex]\( (-1.5, -9.5) \)[/tex]
- The vertical intercept: [tex]\( (0, -5) \)[/tex]
- The coordinates of the two x-intercepts, rounded to two decimal places: [tex]\( (0.68, 0), (-3.68, 0) \)[/tex]
