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Chapter 11: Problem 61

For each of the following, graph the function and find the vertex, the axis of symmetry, the maximum value or the minimum value, and the range of the function. $$ f(x)=2(x+4)^{2}+1 $$

Short Answer

Expert verified
The vertex is \((-4, 1)\), the axis of symmetry is \( x = -4 \), the minimum value is 1, and the range is \( y \geq 1 \).

Step by step solution

01

- Rewrite in Vertex Form

The given function is already in vertex form, which is written as \[ f(x) = a(x-h)^2 + k \]In this case, \( a = 2 \), \( h = -4 \), and \( k = 1 \).
02

- Identify the Vertex

The vertex form of a quadratic function \( f(x) = a(x-h)^2 + k \) gives the vertex directly as the point (h, k). So, the vertex for the function \( f(x) = 2(x+4)^2 + 1 \) is \( (-4, 1) \).
03

- Determine the Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex. It can be found using the x-value of the vertex, which is \( x = -4 \).
04

- Find the Minimum Value

Since \( a = 2 \ and \ a > 0 \), the parabola opens upwards, meaning it has a minimum value at the vertex. The minimum value of the function is the y-value of the vertex, which is \( y = 1 \).
05

- Determine the Range

Because the parabola opens upwards (\( a > 0 \)), the range of the function is all y-values greater than or equal to the minimum value. Therefore, the range is \( y \geq 1 \).
06

- Graph the Function

To graph \( f(x) = 2(x+4)^2 + 1 \), plot the vertex at \( (-4, 1) \), draw the axis of symmetry at \( x = -4 \), and sketch the parabola opening upwards. Make sure to plot additional points on either side of the axis of symmetry to ensure accuracy.

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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 helps us to identify key characteristics of its graph quickly and easily. It is represented as [ f(x) = a(x-h)^2 + k ]. In this form:
  • (a)determines the direction of the parabola (upwards if (a > 0 or downwards if (a < 0)).
  • (h,k)represents the vertex of the parabola, which is its highest or lowest point depending on the direction.
For the function (f(x) = 2(x+4)^2 + 1), we can see both the vertex and the value of 'a' clearly. Here, (a = 2 and (h = -4, k = 1, so the vertex is (-4, 1)and the parabola opens upwards.
Axis of Symmetry
The axis of symmetry of a quadratic function is a vertical line that runs through its vertex. This line divides the parabola into two mirror-image halves, helping in plotting the graph accurately.
For a quadratic function in vertex form (f(x) = a(x-h)^2 + k), the axis of symmetry is (x = h).For our example function (f(x) = 2(x+4)^2 + 1), the axis of symmetry is the vertical line passing through (h = -4), so it's (x = -4).This vertical line helps us when reflecting points across the axis, ensuring our graph maintains its symmetrical nature.
Minimum Value
Quadratic functions in vertex form can either have a minimum or maximum value, depending on whether the parabola opens upwards or downwards. When the parabola opens upwards ((a > 0)), it has a minimum value, which occurs at the vertex.For the function (f(x) = 2(x+4)^2 + 1), a = 2, which means the parabola opens upwards. Consequently, the function achieves its minimum value at the vertex, (-4, 1).Here, the minimum value is the y-coordinate of the vertex, which is (y = 1).This tells us that f(x)is never less than 1.
Range of a Function
The range of a function is the set of all possible y-values that the function can take. For quadratic functions, this depends on whether the parabola opens upwards or downwards. Since our example function (f(x) = 2(x+4)^2 + 1 ) opens upwards (a > 0), the y-values will be greater than or equal to the minimum value.In our function, we determined that the minimum value is 1. Therefore, the range of the function is (y 鈮� 1).This means that no matter what value you put for x, the output (f(x)) will always be 1or greater.

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Most popular questions from this chapter

For each of the following, write the equation of the parabola that has the shape of \(f(x)=2 x^{2}\) or \(g(x)=-2 x^{2}\) and has a maximum value or \(a\) minimum value at the specified point. Minimum: \((2,0)\)

Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ f(x)=x^{2}-10 x-22 $$

Write an equation for a function having a graph with the same shape as the graph of \(f(x)=\frac{3}{5} x^{2},\) but with the given point as the vertex. $$ (4,1) $$

Find (a) the maximum or minimum value and (b) the \(x\) - and \(y\) -intercepts. Round to the nearest hundredth. $$ f(x)=-18.8 x^{2}+7.92 x+6.18 $$

A barge and a fishing boat leave a dock at the same time, traveling at a right angle to each other. The barge travels \(7 \mathrm{km} / \mathrm{h}\) slower than the fishing boat. After 4 hr, the boats are \(68 \mathrm{km}\) apart. Find the speed of each boat. (IMAGE CANNOT COPY)
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