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Chapter 11: Problem 7

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Examples I through 5 . $$ f(x)=(x-5)^{2} $$

Short Answer

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

Step by step solution

01

Identify the Form of the Quadratic Function

The function is given as \( f(x) = (x-5)^2 \), which is in vertex form \( f(x) = a(x-h)^2 + k \). Here, \( a = 1 \), \( h = 5 \), and \( k = 0 \). Since \( a > 0 \), the parabola opens upwards.
02

Determine the Vertex

The vertex of a quadratic function in the form \( (x-h)^2 + k \) is \( (h, k) \). Here, \( h = 5 \) and \( k = 0 \), so the vertex is \( (5, 0) \).
03

Identify the Axis of Symmetry

The axis of symmetry for a parabola \( f(x) = a(x-h)^2 + k \) is the vertical line \( x = h \). Therefore, the axis of symmetry for this function is \( x = 5 \).
04

Sketch the Graph

Draw the parabola opening upwards with the vertex at \( (5, 0) \) on the coordinate plane. The axis of symmetry is a vertical line through the vertex at \( x = 5 \). Mark and label the vertex at \( (5, 0) \) and the axis of symmetry line.

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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 a very convenient way to express a quadratic equation, especially when graphing. It is given by the equation \( f(x) = a(x-h)^2 + k \). Each variable and constant in this equation has a particular meaning:
  • \( a \): The coefficient that indicates the direction and width of the parabola. If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards.
  • \( h \): The x-coordinate of the vertex. It represents the horizontal shift from the origin.
  • \( k \): The y-coordinate of the vertex. It represents the vertical shift from the origin.
Understanding how these parts fit together helps in graphing the parabola quickly and efficiently. For the equation \( f(x) = (x-5)^2 \), it is clear that it is already in vertex form. It shows no vertical shift, as \( k = 0 \), and the parabola opens upwards as \( a = 1 \).
Parabola
A parabola is a U-shaped curve that is the graph of a quadratic function. The general shape can be either opening upwards or downwards. The orientation of a parabola depends on the value of \( a \) in its vertex form equation. Parabolas have several key features:
  • Vertex: The highest or lowest point on the parabola, depending on its direction.
  • Axis of Symmetry: A line that divides the parabola into two mirror-image halves.
  • Direction: Determined by the sign of \( a \); positive or negative.
For the function \( f(x) = (x-5)^2 \), it forms a parabola opening upwards because \( a = 1 \), which is positive. This function illustrates a simple parabola with no stretching or compression, maintaining a standard width.
Vertex and Axis of Symmetry
The vertex of a parabola in vertex form \( f(x) = a(x-h)^2 + k \) is located at \( (h, k) \). For the function \( f(x) = (x-5)^2 \), the vertex is clearly identified as \( (5, 0) \). This point is the minimum point on the parabola since it opens upwards.
The axis of symmetry is a vertical line that passes through the vertex. Its equation is \( x = h \), where \( h \) is the x-coordinate of the vertex. For our example, the axis of symmetry is the line \( x = 5 \).
  • This line divides the parabola into two perfectly symmetrical halves.
  • Each point on one side has a corresponding point on the opposite side of the axis.
Understanding these concepts helps in accurately sketching the graph of a quadratic function and identifying its key features.
Coordinate Plane
The coordinate plane is a two-dimensional space defined by a horizontal axis (x-axis) and a vertical axis (y-axis). It is used to graph equations, like quadratic functions, enabling visualization of shapes like parabolas in relation to their algebraic expressions.
  • x-Axis: The horizontal number line where each point is defined by its x-coordinate.
  • y-Axis: The vertical number line where each point is defined by its y-coordinate.
  • Origin: The point \( (0, 0) \), where the x-axis and y-axis intersect.
For graphing \( f(x) = (x-5)^2 \), plot the vertex at \( (5, 0) \) on the coordinate plane. Then, draw the parabola opening upwards, utilizing the axis of symmetry at \( x = 5 \) to ensure the parabola is symmetrical about this line. The coordinate plane allows you to see how each value of \( x \) is reflected in \( f(x) \).

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Most popular questions from this chapter

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ h(x)=(x-6)^{2}+4 $$

Solve by completing the square. See Section 11.1. $$ x^{2}+14 x+20=0 $$

Add the proper constant to each binomial so that the resulting trinomial is a perfect square trinomial. See Section 11.1. $$ y^{2}+y $$

Notice that the shape of the temperature graph is similar to the curve drawn. In fact, this graph can be modeled by the quadratic function \(f(x)=3 x^{2}-18 x+56,\) where \(f(x)\) is the temperature in degrees Fahrenheit and \(x\) is the number of days from Sunday. (This graph is shown in blue.) Use this function to answer Exercises 85 and 86. Use the function given and the quadratic formula to find when the temperature was \(\left.35^{\circ} \mathrm{F} \text { . [Hint: Let } f(x)=35 \text { and solve for } x .\right]\) Round your answer to one decimal place and interpret your result. Does your answer agree with the graph?

Sketch the graph of each function. See Section 11.5 $$ g(x)=x+2 $$
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