Problem 9 Describe how the graph of \(y=x^... [FREE SOLUTION]

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Discovering Algebra: An Investigative Approach

Jerald Murdock, Ellen Kamischke, Eric Kamischke

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 8: Problem 9

Describe how the graph of $y=x^{2}$ will be transformed if you replace a. $x$ with $(x-3)$ b. $x$ with $(x+2)$ c. $y$ with $(y+2)$ d. $y$ with $(y-3)$

Short Answer

Expert verified

a. Right 3 units, b. Left 2 units, c. Down 2 units, d. Up 3 units.

Step by step solution

Understand Basic Graph Transformations

Before we start, recall that for a standard quadratic function $y = x^2$, a transformation that replaces $x$ with $x-h$ will shift the graph horizontally by $h$ units. A transformation of $y$ by replacing it with $y-k$ will shift the graph vertically by $k$ units.

Transforming by Replacing $x$ with $(x-3)$

Replacing $x$ with $x-3$ in $y = x^2$ gives $y = (x-3)^2$. This is a horizontal shift to the right by 3 units. The vertex of the parabola moves from $(0,0)$ to $(3,0)$.

Transforming by Replacing $x$ with $(x+2)$

Replacing $x$ with $x+2$ results in $y = (x+2)^2$. This is a horizontal shift to the left by 2 units. The vertex moves from $(0,0)$ to $(-2,0)$.

Transforming by Replacing $y$ with $(y+2)$

Substituting $y$ with $y+2$ translates to $y+2 = x^2$ or $y = x^2 - 2$, indicating a vertical shift downwards by 2 units. The vertex changes from $(0,0)$ to $(0,-2)$.

Transforming by Replacing $y$ with $(y-3)$

Replacing $y$ with $y-3$ gives $y-3 = x^2$ or $y = x^2 + 3$, a vertical shift upwards by 3 units. The vertex relocates from $(0,0)$ to $(0,3)$.

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Key Concepts

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

Horizontal Shift

A horizontal shift involves moving the graph of a function left or right along the x-axis. For quadratic functions like $y = x^2$, the transformation is achieved by substituting $x$ with $x-h$, where $h$ is a constant.

If $h$ is positive, the graph shifts to the right. For example, replacing $x$ with $x-3$ results in $y = (x-3)^2$, which shifts the parabola to the right by 3 units.
If $h$ is negative, the graph shifts to the left. Replacing $x$ with $x+2$ changes the function to $y = (x+2)^2$, shifting the graph 2 units to the left.

Understanding horizontal shifts is crucial because it affects the position of the parabola's vertex on the x-axis. These transformations don't alter the shape of the parabola, only its position.

Vertical Shift

Vertical shifts occur when the entire graph of a function moves up or down along the y-axis. This shift is carried out by modifying the $y$-value. For example, in the basic quadratic function $y = x^2$, replacing $y$ with $y-k$ will achieve this effect.

If $k$ is positive, the graph moves upwards. Substituting $y$ with $y-3$ results in $y = x^2 + 3$, which translates the graph up by 3 units.
If $k$ is negative, the graph moves downwards. By replacing $y$ with $y+2$, the equation becomes $y = x^2 - 2$, lowering the graph by 2 units.

Vertical shifts are straightforward but essential for determining the vertical placement of the parabola's vertex. Like horizontal shifts, these only move the graph to a new position without altering its shape.

Parabola Vertex

The vertex of a parabola is a crucial point that determines its form and orientation. For a standard quadratic function $y = x^2$, the vertex sits at the origin $(0,0)$. It moves based on possible horizontal or vertical shifts.
In transformations like $y = (x-3)^2$, the vertex shifts to the right, landing at $(3,0)$. If the transformation is $y = (x+2)^2$, the shift is to the left, placing the vertex at $(-2,0)$.
Vertical transformations also affect the vertex vertically. For instance:

In $y = x^2-2$, the vertex moves to $(0,-2)$, a downward shift by 2 units.
With $y= x^2 + 3$, it relocates to $(0,3)$, moving upwards by 3 units.

Understanding how these shifts affect the vertex helps in sketching the parabola accurately and predicting its behavior on a graph.

Function Transformation

Transformations of functions systematically change their graphs. For quadratic functions, these changes include horizontal and vertical shifts, impacting both position and orientation on a graph.

Horizontal Shifts: These adjust the function left or right along the x-axis, as seen in expressions like $y = (x-h)^2$.
Vertical Shifts: Adjustments up or down along the y-axis, shown in equations such as $y = x^2 + k$.

Knowing these transformations aids in visualizing and graphing quadratic functions accurately. They retain the shape of the parabola but modify the position of its vertex and entire graph, enhancing the understanding of function behavior and graphical representation.

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91影视

Describe how the graph of \(y=x^{2}\) will be transformed if you replace a. \(x\) with \((x-3)\) b. \(x\) with \((x+2)\) c. \(y\) with \((y+2)\) d. \(y\) with \((y-3)\)

Short Answer

Step by step solution

Understand Basic Graph Transformations

Transforming by Replacing \(x\) with \((x-3)\)

Transforming by Replacing \(x\) with \((x+2)\)

Transforming by Replacing \(y\) with \((y+2)\)

Transforming by Replacing \(y\) with \((y-3)\)

Key Concepts

Horizontal Shift

Vertical Shift

Parabola Vertex

Function Transformation

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