Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Ask your questions and receive precise answers from experienced professionals across different disciplines. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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].
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].
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
