At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Let's start with the original function [tex]\( f(x) = x^2 \)[/tex], which is a basic parabola opening upwards with its vertex at the origin [tex]\((0,0)\)[/tex].
The given function is [tex]\( h(x) = -(x-2)^2 \)[/tex]. We will determine the transformations step-by-step:
1. Horizontal Shift:
- In [tex]\( h(x) = -(x-2)^2 \)[/tex], we have [tex]\((x-2)\)[/tex] inside the square term. This represents a horizontal shift.
- Specifically, the graph of [tex]\( f(x) = x^2 \)[/tex] is shifted 2 units to the right.
- Thus, the new function after this shift is [tex]\( g(x) = (x-2)^2 \)[/tex]. The vertex of [tex]\( g(x) \)[/tex] is now at [tex]\((2,0)\)[/tex].
2. Reflection Over the x-axis:
- The negative sign outside the square in [tex]\( h(x) = -(x-2)^2 \)[/tex] causes a reflection over the x-axis.
- Reflecting [tex]\( g(x) = (x-2)^2 \)[/tex] over the x-axis changes the function to [tex]\( h(x) = -(x-2)^2 \)[/tex].
Now, let's look at the transformed function [tex]\( h(x) = -(x-2)^2 \)[/tex]:
- The vertex of the parabola [tex]\( g(x) = (x-2)^2 \)[/tex] at [tex]\((2,0)\)[/tex] is reflected to [tex]\((2,0)\)[/tex] in [tex]\( h(x) = -(x-2)^2 \)[/tex] because reflection preserves the x-coordinate of the vertex, but changes the sign of the y-coordinate.
- This reflection turns the parabola to open downward, and the values of [tex]\( h(x) \)[/tex] will be non-positive.
Let's examine specific values to understand the transformation.
- For [tex]\( x = 0 \)[/tex]:
- Original function [tex]\( f(0) = 0^2 = 0 \)[/tex]
- Horizontal shift: [tex]\( g(0) = (0-2)^2 = 4 \)[/tex]
- Reflection: [tex]\( h(0) = -(0-2)^2 = -4 \)[/tex]
- For [tex]\( x = 1 \)[/tex]:
- Original function [tex]\( f(1) = 1^2 = 1 \)[/tex]
- Horizontal shift: [tex]\( g(1) = (1-2)^2 = 1 \)[/tex]
- Reflection: [tex]\( h(1) = -(1-2)^2 = -1 \)[/tex]
- For [tex]\( x = 2 \)[/tex]:
- Original function [tex]\( f(2) = 2^2 = 4 \)[/tex]
- Horizontal shift: [tex]\( g(2) = (2-2)^2 = 0 \)[/tex]
- Reflection: [tex]\( h(2) = -(2-2)^2 = 0 \)[/tex]
- For [tex]\( x = 3 \)[/tex]:
- Original function [tex]\( f(3) = 3^2 = 9 \)[/tex]
- Horizontal shift: [tex]\( g(3) = (3-2)^2 = 1 \)[/tex]
- Reflection: [tex]\( h(3) = -(3-2)^2 = -1 \)[/tex]
- For [tex]\( x = 4 \)[/tex]:
- Original function [tex]\( f(4) = 4^2 = 16 \)[/tex]
- Horizontal shift: [tex]\( g(4) = (4-2)^2 = 4 \)[/tex]
- Reflection: [tex]\( h(4) = -(4-2)^2 = -4 \)[/tex]
Summarizing the above values:
- [tex]\( h(0) = -4 \)[/tex]
- [tex]\( h(1) = -1 \)[/tex]
- [tex]\( h(2) = 0 \)[/tex]
- [tex]\( h(3) = -1 \)[/tex]
- [tex]\( h(4) = -4 \)[/tex]
Thus, using these transformations, the graph of [tex]\( h(x) = -(x-2)^2 \)[/tex] can be visualized clearly with a vertex at [tex]\((2, 0)\)[/tex], opening downwards. The specific points [tex]\((0, -4), (1, -1), (2, 0), (3, -1), (4, -4)\)[/tex] illustrate the transformations.
The given function is [tex]\( h(x) = -(x-2)^2 \)[/tex]. We will determine the transformations step-by-step:
1. Horizontal Shift:
- In [tex]\( h(x) = -(x-2)^2 \)[/tex], we have [tex]\((x-2)\)[/tex] inside the square term. This represents a horizontal shift.
- Specifically, the graph of [tex]\( f(x) = x^2 \)[/tex] is shifted 2 units to the right.
- Thus, the new function after this shift is [tex]\( g(x) = (x-2)^2 \)[/tex]. The vertex of [tex]\( g(x) \)[/tex] is now at [tex]\((2,0)\)[/tex].
2. Reflection Over the x-axis:
- The negative sign outside the square in [tex]\( h(x) = -(x-2)^2 \)[/tex] causes a reflection over the x-axis.
- Reflecting [tex]\( g(x) = (x-2)^2 \)[/tex] over the x-axis changes the function to [tex]\( h(x) = -(x-2)^2 \)[/tex].
Now, let's look at the transformed function [tex]\( h(x) = -(x-2)^2 \)[/tex]:
- The vertex of the parabola [tex]\( g(x) = (x-2)^2 \)[/tex] at [tex]\((2,0)\)[/tex] is reflected to [tex]\((2,0)\)[/tex] in [tex]\( h(x) = -(x-2)^2 \)[/tex] because reflection preserves the x-coordinate of the vertex, but changes the sign of the y-coordinate.
- This reflection turns the parabola to open downward, and the values of [tex]\( h(x) \)[/tex] will be non-positive.
Let's examine specific values to understand the transformation.
- For [tex]\( x = 0 \)[/tex]:
- Original function [tex]\( f(0) = 0^2 = 0 \)[/tex]
- Horizontal shift: [tex]\( g(0) = (0-2)^2 = 4 \)[/tex]
- Reflection: [tex]\( h(0) = -(0-2)^2 = -4 \)[/tex]
- For [tex]\( x = 1 \)[/tex]:
- Original function [tex]\( f(1) = 1^2 = 1 \)[/tex]
- Horizontal shift: [tex]\( g(1) = (1-2)^2 = 1 \)[/tex]
- Reflection: [tex]\( h(1) = -(1-2)^2 = -1 \)[/tex]
- For [tex]\( x = 2 \)[/tex]:
- Original function [tex]\( f(2) = 2^2 = 4 \)[/tex]
- Horizontal shift: [tex]\( g(2) = (2-2)^2 = 0 \)[/tex]
- Reflection: [tex]\( h(2) = -(2-2)^2 = 0 \)[/tex]
- For [tex]\( x = 3 \)[/tex]:
- Original function [tex]\( f(3) = 3^2 = 9 \)[/tex]
- Horizontal shift: [tex]\( g(3) = (3-2)^2 = 1 \)[/tex]
- Reflection: [tex]\( h(3) = -(3-2)^2 = -1 \)[/tex]
- For [tex]\( x = 4 \)[/tex]:
- Original function [tex]\( f(4) = 4^2 = 16 \)[/tex]
- Horizontal shift: [tex]\( g(4) = (4-2)^2 = 4 \)[/tex]
- Reflection: [tex]\( h(4) = -(4-2)^2 = -4 \)[/tex]
Summarizing the above values:
- [tex]\( h(0) = -4 \)[/tex]
- [tex]\( h(1) = -1 \)[/tex]
- [tex]\( h(2) = 0 \)[/tex]
- [tex]\( h(3) = -1 \)[/tex]
- [tex]\( h(4) = -4 \)[/tex]
Thus, using these transformations, the graph of [tex]\( h(x) = -(x-2)^2 \)[/tex] can be visualized clearly with a vertex at [tex]\((2, 0)\)[/tex], opening downwards. The specific points [tex]\((0, -4), (1, -1), (2, 0), (3, -1), (4, -4)\)[/tex] illustrate the transformations.
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.