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Chapter 8: Problem 10

Describe the order of transformations of the graph of \(y=x^{2}\) represented by a. \(y=-(x+3)^{2}\) (h) b. \(y=0.5(x-2)^{2}+1\)

Short Answer

Expert verified
a. Reflection over x-axis and left 3 units; b. Vertical compression, right 2 units, and up 1 unit.

Step by step solution

01

Step 1a: Identify the base function and transformations

The base function is \( y = x^2 \). For part (a), the function is \( y = -(x+3)^2 \). This equation represents a reflection over the x-axis and a horizontal shift. The minus sign indicates a reflection, while \( (x+3) \) shows a shift to the left by 3 units.
02

Step 2a: Apply transformations to the base function for part a

Starting with \( y = x^2 \), we reflect it over the x-axis to get \( y = -x^2 \). Next, we apply a horizontal shift by replacing \( x \) with \( x+3 \), leading to the transformed equation \( y = -(x+3)^2 \).
03

Step 1b: Identify the base function and transformations for part b

For part (b), the function is \( y = 0.5(x-2)^2 + 1 \). This involves a vertical stretch/compression, a horizontal shift, and a vertical shift. The coefficient \( 0.5 \) indicates a vertical compression, \( (x-2) \) shows a shift to the right by 2 units, and the addition of 1 indicates a vertical shift upwards by 1 unit.
04

Step 2b: Apply transformations to the base function for part b

Starting with \( y = x^2 \), apply the vertical compression by multiplying \( x^2 \) by 0.5 to get \( y = 0.5x^2 \). Next, apply the horizontal shift by replacing \( x \) with \( x-2 \), giving \( y = 0.5(x-2)^2 \). Finally, apply the vertical shift by adding 1, resulting in \( y = 0.5(x-2)^2 + 1 \).

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Key Concepts

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

Quadratic Functions
Quadratic functions are mathematical expressions of the form: \[ y = ax^2 + bx + c \] where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The simplest form is the parent function \( y = x^2 \). When you graph a basic quadratic function, it forms a "U" shaped curve called a parabola. The point where the parabola changes direction is called the vertex, and the line through the vertex, splitting the parabola symmetrically, is known as the axis of symmetry.
  • The graph of a quadratic function opens upwards if \( a > 0 \).
  • If \( a < 0 \), the graph opens downwards and results in a reflection across the x-axis.
  • Complex versions involve transforming the parent graph through various shifts, stretching, or compressing.
Horizontal Shift
A horizontal shift involves moving the entire graph of a function left or right on the Cartesian plane. It changes the x-coordinates of the graph while keeping the y-coordinates the same. In the context of quadratic functions, a horizontal shift is expressed as \[ y = (x - h)^2 \] where \( h \) is the number of units the graph is shifted.
  • To shift the graph to the right, use \( (x - h) \).
  • To shift it to the left, use \( (x + h) \).
For example, in the equation \( y = -(x+3)^2 \), \( (x+3) \) indicates a shift of 3 units to the left. Meanwhile, in \( y = 0.5(x-2)^2 + 1 \), \( (x-2) \) indicates a shift of 2 units to the right.
Vertical Shift
A vertical shift adds or subtracts a constant from the function, moving the graph up or down. This transformation affects the y-values of the graph. For quadratic functions, it is shown as \[ y = x^2 + k \] where \( k \) is the number of units the graph is shifted.
  • Add \( k \): shifts the graph upward by \( k \) units.
  • Subtract \( k \): shifts the graph downward by \( k \) units.
In the equation \( y = 0.5(x-2)^2 + 1 \), adding \( 1 \) indicates that the graph is shifted 1 unit upwards. Vertical shifts, along with horizontal shifts, allow for fine-tuning the position of the graph on the coordinate plane.
Reflection
Reflecting a graph means flipping it over a given axis. In the case of quadratic functions, reflection often occurs over the x-axis. This transformation is expressed by multiplying the entire function by \(-1\). For a quadratic function, a reflection over the x-axis is represented as \[ y = -ax^2 \] This means:
  • The parabola will open downwards instead of upwards.
  • The vertex will remain on the same axis of symmetry, but its y-coordinate will change signs.
An example of a reflected quadratic function is \( y = -(x+3)^2 \), where the minus sign causes the graph to reflect over the x-axis. Reflections are a powerful tool to see the mirror image of a graph across an axis.

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Most popular questions from this chapter

This table lists the vertices of a triangle. Name the vertex or vertices that will not be affected by doing a vertical stretch. (h)

Name the coordinates of the vertex of the graph of \(y=2(x-3)^{2}+1\). Without graphing, name the points on the parabola whose \(x\)-coordinates are 1 unit more or less than the \(x\)-coordinate of the vertex. Check your answers by graphing on your calculator.

Graph Y1 \(1(x)=\operatorname{abs}(x)\) on your calculator. Predict what each graph will look like. Check by comparing the graphs on your calculator. [r] See Calculator Note 8B for specific instructions for your calculator. - ]] a. \(\mathrm{Y}_{2}(x)=-0.5 \mathrm{Y} 1(x)\) b. \(\mathrm{Y} 2(x)=2 \mathrm{Y} 1(x-4)\) c. \(\mathrm{Y}_{2}(x)=-3 \mathrm{Y}_{1}(x+2)+4\)

Graph each function on your calculator. Then describe how each graph relates to the graph of \(y=|x|\) or \(y=x^{2}\). Use the words translation, reflection, vertical stretch, and vertical shrink. a. \(y=2 x^{2}\) b. \(y=0.25|x-2|+1\) (a) c. \(y=-(x+4)^{2}-1\) d. \(y=-2|x-3|+4\)

List L1 and list L2 contain coordinates for three points on the graph of \(f(x)\). List Lz and list LA contain coordinates for the three points after a transformation of \(f\). \begin{tabular}{|c|c|} \hline L1 & L2 \\ \hline \(\boldsymbol{x}\) & \(\boldsymbol{y}\) \\ \hline\(-1\) & 3 \\ \hline 3 & 5 \\ \hline 2 & 4 \\ \hline \end{tabular} \begin{tabular}{|c|c|} \hline \(\mathbf{L 3}\) & L4 \\ \hline\(x\) & \(y\) \\ \hline 7 & \(-1\) \\ \hline 11 & 1 \\ \hline 10 & 0 \\ \hline \end{tabular} a. Write definitions for list L3 and list LA in terms of list \(\mathrm{Ll}_{1}\) and list \(\mathrm{L}_{2}\). b. Describe the transformation.
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