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Chapter 1: Problem 78

How will the graph of \(y=-(x-4)^{2}+8\) differ from the graph of \(y=-x^{2} ?\) Check by graphing both functions together.

Short Answer

Expert verified
The graph shifts 4 units right and 8 units up.

Step by step solution

01

Identify the basic form

The function \( y = -(x-4)^2 + 8 \) is in the vertex form \( y = a(x-h)^2 + k \), where \( a = -1 \), \( h = 4 \), and \( k = 8 \). This indicates that the graph is a parabola opening downwards, shifted from the standard form \( y = -x^2 \).
02

Determine the vertex shift

For \( y = -(x-4)^2 + 8 \), the vertex is at \((h, k)\) which is \((4, 8)\). This means the graph is shifted 4 units to the right and 8 units up from the vertex of \( y = -x^2 \), which is \((0, 0)\).
03

Compare the shape and orientation

Both parabolas have the same shape since they both have \( a = -1 \), meaning they open downwards at the same rate. Therefore, aside from the shift, the two graphs will look identical in terms of width and direction.
04

Graph the functions

Plot the graph of \( y = -x^2 \), which is a downward-opening parabola with vertex at \((0, 0)\). Then plot \( y = -(x-4)^2 + 8 \), which is the same shape but shifted right 4 units and up 8 units to the vertex \((4, 8)\).
05

Verify by checking specific points

Choose specific values for \( x \) to calculate \( y \) for both functions to verify the shifts. For instance, at \( x = 4 \), \( y = -x^2 = -16 \) and \( y = -(x-4)^2 + 8 = 8 \), confirming the vertex shift and shape.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex Form
The vertex form of a quadratic function is a very handy way to express equations and understand parabolas. It is given by the equation \[ y = a(x-h)^2 + k \]where:
  • \(a\): This coefficient affects the width and direction of the parabola. If \(a\) is negative, the parabola opens downwards, and if positive, it opens upwards.
  • \(h\): This is the x-coordinate of the vertex of the parabola. It indicates how far the vertex is shifted horizontally from the origin.
  • \(k\): This is the y-coordinate of the vertex, showing how the vertex is shifted vertically.
These parameters are crucial because changing them alters the position and orientation of the parabola. By identifying \(h\) and \(k\), you quickly know where the vertex, the turning point of the parabola, is located in the coordinate plane. Understanding the vertex form allows you to easily plot or transform a basic parabola.
Vertex Shift
Vertex shift refers to how the vertex of a parabola moves from its position in a standard quadratic form. With the standard equation \( y = ax^2 \), the vertex is at \((0, 0)\). However, when the equation is in vertex form, \( y = a(x-h)^2 + k \), the vertex becomes \((h, k)\).In the exercise provided, for the function \( y = -(x-4)^2 + 8 \), the vertex is at \((4, 8)\). Compare this to the vertex of \( y = -x^2 \), which is at \((0, 0)\). This means the graph of \( y = -(x-4)^2 + 8 \) is shifted:
  • 4 units to the right
  • 8 units upwards
This shift in vertex does not change the general shape or opening direction of the parabola but rather relocates it to a different position in the coordinate plane.
Parabola Orientation
The orientation of a parabola is dictated by the sign of the coefficient "a" in the equation \( y = a(x-h)^2 + k \). This determines whether the parabola opens upwards or downwards:
  • If \(a > 0\), the parabola opens upwards.
  • If \(a < 0\), like in our example, where \(a = -1\), the parabola opens downwards.
Having the same \(a\) value, \( y = -x^2 \) and \( y = -(x-4)^2 + 8 \) both open downwards. This means the y-values decrease as you move away from the vertex along the x-axis.This orientation also implies the parabola is concave, forming a ""cup turned upside down"" shape.
Function Plotting
Plotting these functions on a graph brings out more than just visual validation鈥攊t's a way to see how algebra translates to geometry. To plot \( y = -x^2 \),which forms a symmetric parabola around the y-axis, place the vertex at \((0, 0)\).For \( y = -(x-4)^2 + 8 \),the steps are:
  • Start by marking the vertex at \((4, 8)\).
  • From there, utilize symmetry. Move equidistant points on either side of the vertex along the x-axis to outline the parabola.
  • Remember the same decreasing pattern as \( y = -x^2 \) since the "a" value is unchanged.
Graphing both functions together will distinctly show the vertex shift and how both parabolas with identical shapes still occupy separate positions in the graph. Try plotting a few specific points like \( x = 2 \) and \( x = 6 \) to check for symmetry and correctness.

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