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Chapter 9: Problem 305

Graph each function using a horizontal shift. $$f(x)=(x+5)^{2}$$

Short Answer

Expert verified
Shift the graph of \(f(x) = x^2\) left by 5 units; the new vertex is at (-5,0).

Step by step solution

01

Understand the Parent Function

The parent function is the basic form of a function before any transformations are applied. In this case, the parent function is \(f(x) = x^2\). This is a standard quadratic function.
02

Identify the Transformation

Notice the function given is \(f(x) = (x + 5)^2\). The \((x + 5)\) denotes a horizontal shift. In a function \(f(x + h)\), where \(h\) is a positive number, the graph shifts to the left by \(h\) units. So, in this case, the graph shifts to the left by 5 units.
03

Transform the Vertex

The vertex of the parent function \(f(x) = x^2\) is at (0,0). Applying the horizontal shift to this vertex, we move 5 units to the left. The new vertex will be (-5,0).
04

Sketch the Graph

Draw the graph of the parent function \(f(x) = x^2\). Then shift the entire graph 5 units to the left. The new graph should have its vertex at (-5,0) and will look identical in shape to \(f(x) = x^2\), but it is shifted left.

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Key Concepts

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

parent function
A parent function is the simplest form of a given type of function. It serves as the foundation upon which all transformations are based. For quadratic functions, the parent function is \(f(x) = x^2\). This quadratic function creates a standard U-shaped curve called a parabola with its vertex at the origin (0,0). All other quadratic functions can be seen as transformations of this basic shape.
quadratic function
A quadratic function is a polynomial function of degree 2. Its general form is \(f(x) = ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants, and \(a eq 0\). In this context, the quadratic function \(f(x) = (x + 5)^2\) starts with the parent function and applies transformations to it. The vertex form \(f(x) = a(x-h)^2 + k\) is particularly useful for identifying the transformations applied to the parent function. Here, \(a\) determines the width and direction of the parabola, \(-h\) indicates a horizontal shift, and \(k\) indicates a vertical shift.
vertex transformation
Vertex transformation, particularly in quadratic functions, involves moving the position of the parabola's vertex. In the function \(f(x) = (x+5)^2\), the term \(x+5\) results in a horizontal shift. In general, for \(f(x + h)\), the graph shifts left by \(h\) units if \(h\) is positive, and right if \(h\) is negative. Here, the vertex of the parent function \(f(x) = x^2\) is at (0,0). Applying the horizontal shift of 5 units left results in a new vertex at (-5,0).
graphing transformations
Graphing transformations of quadratic functions can involve translating, stretching, compressing, and reflecting the graph.
  • Horizontal Shift: Moving the graph left or right.
  • Vertical Shift: Moving the graph up or down.
  • Reflection: Flipping the graph over the x-axis or y-axis.
  • Stretching/Compressing: Changing the width or height of the graph.
For the function \(f(x) = (x+5)^2\), we only applied a horizontal shift. Starting with the parent function \(f(x) = x^2\), we moved the graph 5 units to the left. This horizontal shift changed the vertex to (-5,0), but the shape of the parabola stayed the same. Remember that each type of transformation changes only specific aspects of the graph without affecting its overall shape.

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Most popular questions from this chapter

Rewrite each function in the \(f(x)=a(x-h)^{2}+k\) form by completing the square. $$f(x)=-4 x^{2}-16 x-9$$

Graph each function. $$f(x)=\frac{1}{4} x^{2}$$

Graph each function using a vertical shift. $$g(x)=x^{2}+5$$

Graph each function. $$f(x)=-4 x^{2}$$

Find the intercepts of the parabola whose function is given. $$f(x)=-x^{2}-6 x-9$$
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