Let's work step-by-step to find the necessary values for the given quadratic function [tex]\( f(x) = 2x^2 + 12x + 9 \)[/tex].
### 1. Finding the [tex]\( y \)[/tex]-intercept
The [tex]\( y \)[/tex]-intercept of a function is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
So, we need to evaluate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 2(0)^2 + 12(0) + 9 = 9 \][/tex]
Thus, the [tex]\( y \)[/tex]-intercept is:
[tex]\[ y = 9 \][/tex]
### 2. Finding the [tex]\( x \)[/tex]-intercepts
The [tex]\( x \)[/tex]-intercepts of the function are the values of [tex]\( x \)[/tex] where [tex]\( f(x) = 0 \)[/tex].
So, we need to solve the equation:
[tex]\[ 2x^2 + 12x + 9 = 0 \][/tex]
By solving this quadratic equation, we get the [tex]\( x \)[/tex]-intercepts:
[tex]\[ x_1 = -3 - \frac{3\sqrt{2}}{2} \][/tex]
[tex]\[ x_2 = -3 + \frac{3\sqrt{2}}{2} \][/tex]
### 3. Finding the Turning Point (Vertex)
The turning point of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found using the vertex formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For our function [tex]\( f(x) = 2x^2 + 12x + 9 \)[/tex]:
[tex]\[ a = 2, \quad b = 12 \][/tex]
So,
[tex]\[ x = -\frac{12}{2 \cdot 2} = -3 \][/tex]
To find the corresponding [tex]\( y \)[/tex]-coordinate of the vertex:
[tex]\[ f(-3) = 2(-3)^2 + 12(-3) + 9 = 18 - 36 + 9 = -9 \][/tex]
Thus, the turning point (vertex) is:
[tex]\[ (-3, -9) \][/tex]
### Summary
- The [tex]\( y \)[/tex]-intercept is [tex]\( 9 \)[/tex].
- The [tex]\( x \)[/tex]-intercepts are [tex]\( -3 - \frac{3\sqrt{2}}{2} \)[/tex] and [tex]\( -3 + \frac{3\sqrt{2}}{2} \)[/tex].
- The turning point (vertex) of the function is [tex]\( (-3, -9) \)[/tex].
