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Chapter 8: Problem 23

Graph the function, label the vertex, and draw the axis of symmetry. $$ f(x)=2(x+1)^{2} $$

Short Answer

Expert verified
The vertex is at \( (-1, 0) \) and the axis of symmetry is \( x = -1 \).

Step by step solution

01

Identify the Form of the Function

The given function is in the form of a quadratic function: \[ f(x) = a(x-h)^2 + k \] where \( a = 2 \), \( h = -1 \) (since it is \( x+1 \)), and \( k = 0 \). This helps identify the vertex and the direction of the parabola.
02

Find the Vertex

In the standard form \( f(x) = a(x-h)^2 + k \), the vertex is at point \( (h, k) \). Here, \( h = -1 \) and \( k = 0 \). So, the vertex of the function is at \( (-1, 0) \).
03

Determine the Axis of Symmetry

The axis of symmetry of a parabola in the form \( f(x) = a(x-h)^2 + k \) is a vertical line that passes through the vertex. It is given by \( x = h \). Here, \( h = -1 \). Therefore, the axis of symmetry is \( x = -1 \).
04

Graph the Function

To graph the function, plot the vertex at \( (-1, 0) \) and draw the parabola opening upward since \( a = 2 \) is positive. The function will be symmetric around the axis \( x = -1 \). For graphing, choose additional points if needed, such as \( x = -2 \) and \( x = 0 \) and calculate their corresponding \( y \)-values.
05

Label the Vertex and Draw the Axis of Symmetry

On the graph, clearly label the vertex at \( (-1, 0) \) and draw the axis of symmetry as a dashed vertical line through \( x = -1 \).

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Key Concepts

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

Vertex
The vertex of a quadratic function is the highest or lowest point on its graph, known as the parabola. For the function given, \[ f(x) = 2(x+1)^2 \] we use the form \( f(x) = a(x-h)^2 + k \), where \( h \) and \( k \) determine the vertex. Here, we identify that \( h = -1 \) and \( k = 0 \). Thus, the vertex is at \( (-1, 0) \).
This point helps in determining the shape and position of the parabola.
Always label the vertex clearly on the graph to show the maximum or minimum value of the function.
Axis of Symmetry
The axis of symmetry is a vertical line that splits the parabola into two mirror images. For a quadratic function in the form \( f(x) = a(x-h)^2 + k \), the axis of symmetry is given by the equation \( x = h \).
In our function, \( h = -1 \), so the axis of symmetry is \( x = -1 \).
This line is important because it helps in graphing the quadratic function by ensuring that the points on the graph are symmetric around this line.
Always represent the axis of symmetry with a dashed line on your graph for clear understanding and accuracy.
Parabola Opening
The direction in which a parabola opens is determined by the coefficient \( a \) in the quadratic function \( f(x) = a(x-h)^2 + k \).
If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.
For the given function, \( a = 2 \), which is positive, meaning the parabola opens upward.
This affects the general shape of the graph and can help in plotting additional points which will always lie above the vertex for an upward opening parabola.
Monitoring the sign of \( a \) is crucial for identifying whether the function represents a minimum or a maximum point at the vertex.

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