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Chapter 3: Problem 48

(a) Complete the given table. $$\begin{array}{l}x \quad x^{2} \quad(x-1)^{2} \quad(x+1)^{2} \\\\\hline 0 \\\1 \\\2 \\\3 \\\\-1 \\\\-2 \\\\-3 \\\\\hline\end{array}$$ (b) Using the results in the table, graph the functions \(y=x^{2}, y=(x-1)^{2},\) and \(y=(x+1)^{2}\) on the same set of axes. How are the graphs related?

Short Answer

Expert verified
Graphs are parabolas; \((x-1)^2\) is shifted right, \((x+1)^2\) is shifted left of \(x^2\).

Step by step solution

01

Calculate \(x^2\)

Substitute each value of \(x\) into the equation \(y = x^2\) to find the square of \(x\). The results are: - If \(x = 0\), then \(x^2 = 0^2 = 0\).- If \(x = 1\), then \(x^2 = 1^2 = 1\).- If \(x = 2\), then \(x^2 = 2^2 = 4\).- If \(x = 3\), then \(x^2 = 3^2 = 9\).- If \(x = -1\), then \(x^2 = (-1)^2 = 1\).- If \(x = -2\), then \(x^2 = (-2)^2 = 4\).- If \(x = -3\), then \(x^2 = (-3)^2 = 9\).
02

Calculate \((x-1)^2\)

Substitute each value of \(x\) into the expression \((x-1)^2\) and calculate:- If \(x = 0\), then \((x-1)^2 = (0-1)^2 = 1\).- If \(x = 1\), then \((x-1)^2 = (1-1)^2 = 0\).- If \(x = 2\), then \((x-1)^2 = (2-1)^2 = 1\).- If \(x = 3\), then \((x-1)^2 = (3-1)^2 = 4\).- If \(x = -1\), then \((x-1)^2 = (-1-1)^2 = 4\).- If \(x = -2\), then \((x-1)^2 = (-2-1)^2 = 9\).- If \(x = -3\), then \((x-1)^2 = (-3-1)^2 = 16\).
03

Calculate \((x+1)^2\)

Substitute each value of \(x\) into the expression \((x+1)^2\) and calculate:- If \(x = 0\), then \((x+1)^2 = (0+1)^2 = 1\).- If \(x = 1\), then \((x+1)^2 = (1+1)^2 = 4\).- If \(x = 2\), then \((x+1)^2 = (2+1)^2 = 9\).- If \(x = 3\), then \((x+1)^2 = (3+1)^2 = 16\).- If \(x = -1\), then \((x+1)^2 = (-1+1)^2 = 0\).- If \(x = -2\), then \((x+1)^2 = (-2+1)^2 = 1\).- If \(x = -3\), then \((x+1)^2 = (-3+1)^2 = 4\).
04

Fill in the Complete Table

Using the results from Steps 1, 2, and 3, fill in the table:\[\begin{array}{c|c|c|c}x & x^2 & (x-1)^2 & (x+1)^2 \hline0 & 0 & 1 & 1 \1 & 1 & 0 & 4 \2 & 4 & 1 & 9 \3 & 9 & 4 & 16 \-1 & 1 & 4 & 0 \-2 & 4 & 9 & 1 \-3 & 9 & 16 & 4 \hline\end{array}\]
05

Graph the Functions

Plot the points for each function based on the completed table from Step 4 on the same set of axes: - The graph of \(y = x^2\) is centered at the origin and symmetrical about the y-axis.- The graph of \(y = (x-1)^2\) is a parabola shifted one unit to the right compared to \(y = x^2\).- The graph of \(y = (x+1)^2\) is a parabola shifted one unit to the left compared to \(y = x^2\).
06

Analyze the Graphs' Relationships

Observe that the graph of \(y = x^2\) is the standard parabola. The graphs of \(y = (x-1)^2\) and \(y = (x+1)^2\) are horizontally shifted versions of \(y = x^2\), with \(y = (x-1)^2\) shifted right by one unit, and \(y = (x+1)^2\) shifted left by one unit.

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Key Concepts

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

Parabolas
In mathematics, a parabola is a symmetrical, U-shaped curve that is the graph of a quadratic function. The standard form of a quadratic function is given by the equation \(y = ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants. When you run into the simplest quadratic function, \(y = x^2\), the resulting graph is a parabola which is symmetric about the y-axis.

Parabolas have several key characteristics:
  • They have a vertex, which is the highest or lowest point on the graph, depending on the orientation of the parabola.
  • They have a line of symmetry that passes through the vertex, making one side of the graph a mirror image of the other.
  • The direction they open is determined by the sign of the leading coefficient \(a\); if \(a\) is positive, the parabola opens upwards. If \(a\) is negative, it opens downwards.
Understanding these properties helps in analyzing changes and transformations of parabolas within quadratic functions.
Function Transformation
Function transformation involves shifting, stretching, compressing or other modifications of functions. When working with quadratic functions, one common transformation is translating the parabola along the x-axis or the y-axis.

In the example, the functions \((x-1)^2\) and \((x+1)^2\) demonstrate horizontal shifts:
  • \(y = (x-1)^2\) shifts the parabola one unit to the right. This is because every \(x\) value now needs to be increased by one to yield the same y-value as \(y = x^2\).
  • \(y = (x+1)^2\) shifts the parabola one unit to the left. Here, each x-value is decreased by one for the same effect as \(y = x^2\).
These transformations do not change the shape of the parabola, only its position. Recognizing these shifts is essential for graphing and understanding how adjustments to the equation affect the graph.
Graphing Functions
Graphing functions is a powerful way to understand the behavior and properties of equations. For quadratic functions, the graph is a parabola. To graph these functions:
  • Start by plotting key points, such as the vertex and intercepts, based on the function equation.
  • Consider the direction and width of the parabola as defined by the coefficients.
  • Use symmetry to plot additional points, ensuring the curve is symmetrical around the line of symmetry.
In our exercise, comparing the graphs of \(y = x^2\), \(y = (x-1)^2\), and \(y = (x+1)^2\) on the same axes reveals their relationships:
  • All three graphs are parabolas, with the same shape and orientation.
  • They demonstrate horizontal shifts, with no changes in the vertical stretch or compression.
  • Understanding these similarities and differences will aid in predicting how changes in a quadratic equation affect its graph.
Graphing requires a good grasp of transformations and symmetry, making it easier to see the effects of different alterations on the equation's graph.

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Most popular questions from this chapter

Let the function \(L\) be defined by the following rule: \(L(x)\) is the exponent to which 2 must be raised to yield \(x\). (For the moment, we won't concern ourselves with the domain and range.) Then \(L(8)=3,\) for example, since the exponent to which 2 must be raised to yield 8 is 3 (that is, \(8=2^{3}\) ). Find the following outputs (a) \(L(1)\) (b) \(L(2)\) (c) \(L(4)\) (d) \(L(64)\) (e) \(L(1 / 2)\) (f) \(L(1 / 4)\) (g) \(L(1 / 64)\) (h) \(L(\sqrt{2})\)

$$\text { Let } f(z)=\frac{3 z-4}{5 z-3} . \text { Find } f\left(\frac{3 z-4}{5 z-3}\right)$$

For Exercises 58 and \(59,\) assume that \((a, b)\) is a point on the graph of \(y=f(x),\) and specify the corresponding point on the graph of each equation. [For example, the point that corre- sponds to \((a, b)\) on the graph of \(y=f(x-1)\) is (\(a+1, b).] (a) \)y=f(x-3)\( (b) \)y=f(x)-3\( (c) \)y=f(x-3)-3\( (d) \)y=-f(x)$

Let \(F(x)=-x^{2}\) and \(G(x)=\sqrt{x} .\) Determine the domains of \(F \circ G\) and \(G \circ F\).

Use a graphing utility to graph each function and then apply the horizontal line test to see whether the function is one-to-one. $$y=2 x^{5}+x-1$$
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