Let's consider the quadratic function [tex]\( f(x) = -x^2 - 2x + 24 \)[/tex].
1. Finding the [tex]\( x \)[/tex]-intercepts:
The [tex]\( x \)[/tex]-intercepts are the points where the function [tex]\( f(x) \)[/tex] intersects the [tex]\( x \)[/tex]-axis. These occur where [tex]\( f(x) = 0 \)[/tex]. Thus, we need to solve the equation:
[tex]\[
-x^2 - 2x + 24 = 0
\][/tex]
Solving this quadratic equation, we find the [tex]\( x \)[/tex]-intercepts to be:
[tex]\[
x = -6 \quad \text{and} \quad x = 4
\][/tex]
Therefore, the [tex]\( x \)[/tex]-intercepts are [tex]\( (-6, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex].
2. Finding the [tex]\( y \)[/tex]-intercept:
The [tex]\( y \)[/tex]-intercept is the point where the function [tex]\( f(x) \)[/tex] intersects the [tex]\( y \)[/tex]-axis. This occurs when [tex]\( x = 0 \)[/tex]. So, we evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = -0^2 - 2(0) + 24 = 24
\][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is [tex]\( (0, 24) \)[/tex].
3. Finding the vertex:
The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula for the [tex]\( x \)[/tex]-coordinate of the vertex:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
For the given function [tex]\( f(x) = -x^2 - 2x + 24 \)[/tex], the coefficients are [tex]\( a = -1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 24 \)[/tex]. Substituting these values into the formula:
[tex]\[
x = -\frac{-2}{2(-1)} = 1
\][/tex]
Now, we find the corresponding [tex]\( y \)[/tex]-coordinate by evaluating [tex]\( f(x) \)[/tex] at this [tex]\( x \)[/tex]-value:
[tex]\[
f(-1) = -(-1)^2 - 2(-1) + 24 = -1 + 2 + 24 = 25
\][/tex]
Therefore, the vertex is [tex]\( (-1, 25) \)[/tex].
4. Finding the line of symmetry:
The line of symmetry in a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by the line [tex]\( x = -\frac{b}{2a} \)[/tex]. As we found earlier:
[tex]\[
x = -\frac{-2}{2(-1)} = 1
\][/tex]
Therefore, the line of symmetry of the given quadratic function is [tex]\( x = -1 \)[/tex].
Summarizing:
- The [tex]\( x \)[/tex]-intercepts are [tex]\( (-6, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex].
- The [tex]\( y \)[/tex]-intercept is [tex]\( (0, 24) \)[/tex].
- The vertex is [tex]\( (-1, 25) \)[/tex].
- The line of symmetry is [tex]\( x = -1 \)[/tex].
