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Chapter 11: Problem 4

Graph each quadratic function. Label the vertex and sketch and label the axis of symmetry. \(g(x)=(x+5)^{2}\)

Short Answer

Expert verified
Vertex: (-5, 0); Axis of symmetry: x = -5.

Step by step solution

01

Identify the basic form of the quadratic function

The given function is \(g(x) = (x + 5)^2\). This quadratic function is in the vertex form \(f(x) = (x - h)^2 + k\), where \(h\) is the x-coordinate of the vertex and \(k\) is the y-coordinate. By comparing \(g(x) = (x + 5)^2\) with \(f(x) = (x - h)^2 + k\), we can see that \(h = -5\) and \(k = 0\).
02

Identify the vertex

The vertex can be directly identified from the vertex form of the function. Since \(h = -5\) and \(k = 0\), the vertex is at \((-5, 0)\).
03

Sketch the graph

Draw the axis of the graph and plot the vertex at \((-5, 0)\). The quadratic function \(g(x) = (x + 5)^2\) is a parabola that opens upwards, as the coefficient of \((x + 5)^2\) is positive. Plot additional points if needed, such as \((-4,1)\) and \((-6,1)\), to help draw the curve.
04

Label the axis of symmetry

The axis of symmetry for a quadratic function in vertex form \((x - h)^2\) is the vertical line \(x = h\). For \(g(x) = (x + 5)^2\), the axis of symmetry is \(x = -5\). Draw and label this line on the graph.

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Key Concepts

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

Vertex Form
The vertex form of a quadratic function is an effective way to represent a parabola. It makes identifying key elements such as the vertex straightforward. The general expression for the vertex form is \(f(x) = (x - h)^2 + k\). Here, \(h\) and \(k\) give you the coordinates of the vertex, which is a crucial point on the graph of the function.
  • Vertex: The vertex is where the graph changes direction. In a quadratic \((x - h)^2 + k\), the vertex is \((h, k)\).
  • Easy Graphing: By identifying \(h\) and \(k\), you can easily plot the main point. From there, sketch the curve around this key point.
For the function \(g(x) = (x + 5)^2\), rewriting it as \(f(x) = (x - (-5))^2 + 0\) shows its vertex. In this case, \(h = -5\) and \(k = 0\), which means the vertex is at the point \((-5, 0)\). This point becomes the central reference for sketching the parabola.
Axis of Symmetry
The axis of symmetry is essential to understanding the balance of a parabola. It is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. In vertex form, the axis of symmetry can be easily identified.
  • Defining the Line: Given a quadratic in vertex form \(f(x) = (x - h)^2 + k\), the axis of symmetry is the line \(x = h\).
  • Purpose: This line ensures that for every point on one side of the axis, there is a corresponding point on the opposite side.
For the function \(g(x) = (x + 5)^2\), this translates to an axis of symmetry at \(x = -5\). This line is critical for drawing the parabola as it helps ensure the curve is symmetrical. When sketching, you can draw a dashed vertical line at \(x = -5\) as a guide in plotting the graph.
Parabola
A parabola is a U-shaped curve representing a quadratic function. It has distinct features which include the vertex and the axis of symmetry. Understanding the nature of a parabola helps in accurately graphing quadratic functions.
  • Shape & Direction: Parabolas can open upwards or downwards. An upward opening occurs when the coefficient of \((x - h)^2\) is positive in the vertex form. Conversely, it opens downwards if the coefficient is negative.
  • Points to Consider: Besides the vertex, selecting other strategic points on the graph helps anchor the parabolic shape. In the case of \(g(x) = (x + 5)^2\), points such as \((-4, 1)\) and \((-6, 1)\) verify that the graph is widening symmetrically around the axis of symmetry.
The quadratic function \(g(x) = (x+5)^2\) forms a parabola that opens upwards, with its main features being symmetric about \(x = -5\). Knowing these attributes allows a more intuitive and precise sketch of the function on a graph.

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Most popular questions from this chapter

Use the quadratic formula to solve each equation. These equations have real number solutions only. $$ (m+2)(2 m-6)=5(m-1)-12 $$

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