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To determine the graph of the given function [tex]\( h(x) = -(x+5)^2 \)[/tex] through transformations of the parent function [tex]\( f(x) = x^2 \)[/tex], follow these steps:
1. Start with the parent function [tex]\( f(x) = x^2 \)[/tex]:
This is a basic parabola that opens upwards, with its vertex at the origin [tex]\((0, 0)\)[/tex].
2. Apply a horizontal shift to the left by 5 units:
To shift the graph of [tex]\( f(x) = x^2 \)[/tex] horizontally to the left by 5 units, replace [tex]\( x \)[/tex] with [tex]\( x + 5 \)[/tex]. This gives us the function [tex]\( g(x) = (x + 5)^2 \)[/tex].
The graph of [tex]\( g(x) = (x + 5)^2 \)[/tex] is still a parabola that opens upwards, but its vertex is now shifted to the point [tex]\((-5, 0)\)[/tex].
3. Reflect across the x-axis:
To reflect the graph of [tex]\( g(x) = (x + 5)^2 \)[/tex] across the x-axis, multiply the function by [tex]\(-1\)[/tex]. This transformation gives us [tex]\( h(x) = -(x + 5)^2 \)[/tex].
The graph of [tex]\( h(x) = -(x + 5)^2 \)[/tex] is a parabola that opens downwards, and its vertex remains at the point [tex]\((-5, 0)\)[/tex].
4. Verification of points:
We can calculate specific points to illustrate how the transformations affect some key values. Consider the x-values [tex]\(-6, -5, -4, 0, 4\)[/tex] and determine the corresponding y-values for [tex]\( h(x) = -(x+5)^2 \)[/tex]:
- For [tex]\( x = -6 \)[/tex]:
[tex]\[ h(-6) = -(-6 + 5)^2 = -1^2 = -1 \][/tex]
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ h(-5) = -(-5 + 5)^2 = -0^2 = 0 \][/tex]
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ h(-4) = -(-4 + 5)^2 = -1^2 = -1 \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = -(0 + 5)^2 = -5^2 = -25 \][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ h(4) = -(4 + 5)^2 = -9^2 = -81 \][/tex]
So, the specific points on the graph of [tex]\( h(x) = -(x+5)^2 \)[/tex] are as follows:
[tex]\[ (-6, -1), (-5, 0), (-4, -1), (0, -25), (4, -81) \][/tex]
These points confirm the transformations applied to the parent function [tex]\( f(x) = x^2 \)[/tex]. The graph of [tex]\( h(x) = -(x+5)^2 \)[/tex] is a downward-opening parabola with its vertex at [tex]\((-5, 0)\)[/tex], and it is horizontally shifted 5 units to the left and reflected across the x-axis from the graph of [tex]\( f(x) = x^2 \)[/tex].
1. Start with the parent function [tex]\( f(x) = x^2 \)[/tex]:
This is a basic parabola that opens upwards, with its vertex at the origin [tex]\((0, 0)\)[/tex].
2. Apply a horizontal shift to the left by 5 units:
To shift the graph of [tex]\( f(x) = x^2 \)[/tex] horizontally to the left by 5 units, replace [tex]\( x \)[/tex] with [tex]\( x + 5 \)[/tex]. This gives us the function [tex]\( g(x) = (x + 5)^2 \)[/tex].
The graph of [tex]\( g(x) = (x + 5)^2 \)[/tex] is still a parabola that opens upwards, but its vertex is now shifted to the point [tex]\((-5, 0)\)[/tex].
3. Reflect across the x-axis:
To reflect the graph of [tex]\( g(x) = (x + 5)^2 \)[/tex] across the x-axis, multiply the function by [tex]\(-1\)[/tex]. This transformation gives us [tex]\( h(x) = -(x + 5)^2 \)[/tex].
The graph of [tex]\( h(x) = -(x + 5)^2 \)[/tex] is a parabola that opens downwards, and its vertex remains at the point [tex]\((-5, 0)\)[/tex].
4. Verification of points:
We can calculate specific points to illustrate how the transformations affect some key values. Consider the x-values [tex]\(-6, -5, -4, 0, 4\)[/tex] and determine the corresponding y-values for [tex]\( h(x) = -(x+5)^2 \)[/tex]:
- For [tex]\( x = -6 \)[/tex]:
[tex]\[ h(-6) = -(-6 + 5)^2 = -1^2 = -1 \][/tex]
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ h(-5) = -(-5 + 5)^2 = -0^2 = 0 \][/tex]
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ h(-4) = -(-4 + 5)^2 = -1^2 = -1 \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = -(0 + 5)^2 = -5^2 = -25 \][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ h(4) = -(4 + 5)^2 = -9^2 = -81 \][/tex]
So, the specific points on the graph of [tex]\( h(x) = -(x+5)^2 \)[/tex] are as follows:
[tex]\[ (-6, -1), (-5, 0), (-4, -1), (0, -25), (4, -81) \][/tex]
These points confirm the transformations applied to the parent function [tex]\( f(x) = x^2 \)[/tex]. The graph of [tex]\( h(x) = -(x+5)^2 \)[/tex] is a downward-opening parabola with its vertex at [tex]\((-5, 0)\)[/tex], and it is horizontally shifted 5 units to the left and reflected across the x-axis from the graph of [tex]\( f(x) = x^2 \)[/tex].
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