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Chapter 10: Problem 27

For Exercises \(24-27,\) use the equation \(x=3 y^{2}+4 y+1\) How does the graph compare to the graph of the parent function \(x=y^{2} ?\)

Short Answer

Expert verified
The graph of \(x = 3y^2 + 4y + 1\) is horizontally wider, shifted right, and skewed compared to \(x = y^2\).

Step by step solution

01

Understanding the Parent Function

The parent function is given by the equation \( x = y^2 \). This represents a parabola that opens to the right on the Cartesian plane, with its vertex at the origin (0,0). Each point \((x,y)\) on the parabola satisfies \( x = y^2 \).
02

Identifying the Given Function

The given function is \( x = 3y^2 + 4y + 1 \). This also represents a parabola in the \(xy\)-plane, but it includes additional terms compared to the parent function, which will affect its shape, position, and orientation.
03

Analyzing the Effect of the Coefficient of \(y^2\)

The coefficient of \(y^2\) in the given function is 3, compared to 1 in the parent function. This will make the graph of the given function \'wider\' than the parent function because it stretches the parabola along the \(x\)-direction.
04

Considering the Linear Term \(4y\)

The term \(4y\) indicates a linear shift and results in the parabola being skewed. Such a term can shift the vertex along the \(y\)-axis and change the symmetry of the graph.
05

Impact of the Constant Term \(+1\)

The term \(+1\) shifts the entire graph of the parabola 1 unit to the right along the \(x\)-axis. This affects where the vertex lies in the \(xy\)-plane relative to the origin.
06

Comparing the Graphs

Compared to the parent function \(x = y^2\), the graph of \(x = 3y^2 + 4y + 1\) is stretched horizontally, shifted in the \(x\)-axis, and skewed with an asymmetrical shape. The vertex is no longer at the origin.

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Key Concepts

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

Parent Function
In mathematics, the term "parent function" refers to the simplest form of a given family of functions. It acts as the foundational template from which more complex functions can be derived. Here, the parent function is represented by the equation \(x = y^2\). This equation defines a parabola that opens to the right on a Cartesian plane.

Key properties of this parent function include:
  • The graph is symmetrical about the x-axis.
  • The vertex is located at the origin (0,0).
  • As the variable \(y\) increases or decreases, \(x\) only takes non-negative values.
Understanding the parent function is crucial for identifying how transformations and additional terms alter the graph's properties and position.
Coefficients
In the context of equations, coefficients are numerical values that multiply a variable. In our exercise, the parent function has a coefficient of 1 in front of \(y^2\), while the given function \(x = 3y^2 + 4y + 1\) has a coefficient of 3. This coefficient plays a significant role in transforming the graph.

Key effects of the coefficient of \(y^2\):
  • A larger coefficient (like 3) results in a horizontal stretch of the parabola, making it open more widely compared to the parent function.
  • This transformation can be visualized as 'flattening' the graph along the x-axis, indicating a faster spread of x-values for small changes in y.
Understanding the impact of coefficients helps us see how each term in the equation modifies the shape and orientation of the graph.
Graph Transformations
Graph transformations are changes made to a graph's appearance by altering the function's equation. In our example, these transformations occur due to the linear and constant terms in the function \(x = 3y^2 + 4y + 1\).

Types of transformations include:
  • Horizontal Stretch: Due to the coefficient of 3 in 3y^2, the graph expands horizontally compared to the parent function.
  • Linear Shift: The 4y term introduces a shearing transformation, altering the symmetry typically associated with parabolas and affecting the parabola's lopsidedness.
  • Horizontal Shift: The +1 constant term shifts the whole parabola one unit to the right.
These transformations illustrate how each component of a function's equation uniquely contributes to altering the graph's final appearance.
Vertex Shift
The concept of vertex shift refers to the movement of a parabola's vertex due to changes in the equation. For the given function, the vertex no longer sits at the origin.

Factors causing vertex shift:
  • The presence of the linear term \(4y\) skews the graph, which displaces the original vertex.
  • The constant term \(+1\) in the function contributes to moving the vertex along the x-axis.
Calculating the new vertex requires examining how each term adjusts the graph's key point. These shifts are important because they determine where the lowest or highest point (vertex) of the parabola is located on the graph. Such insights are invaluable for graph interpretation and real-world applications based upon parabolic equations.

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Most popular questions from this chapter

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{y^{2}=x^{2}-7} \\ {x^{2}+y^{2}=25}\end{array} $$

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{y+x^{2}=3} \\ {x^{2}+4 y^{2}=36}\end{array} $$

Graph the line with the given equation. \(y=-2 x\)

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{3 x=8 y^{2}} \\ {8 y^{2}-2 x^{2}=16}\end{array} $$

Each equation is of the form \(A x^{2}+B x y+C y^{2}+D x+\) \(E y+F=0 .\) Identify the values of \(A, B,\) and \(C\). $$ -x y-2 x-3 y+6=0 $$
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