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Chapter 8: Problem 46

Graph the function and find the vertex, the axis of symmetry, and the maximum value or the minimum value. $$ g(x)=\frac{3}{2}(x+2)^{2}-1 $$

Short Answer

Expert verified
Vertex: (-2, -1); Axis of Symmetry: x = -2; Minimum value: -1;

Step by step solution

01

- Identify the form of the quadratic function

The given function is \[ g(x) = \frac{3}{2}(x+2)^{2}-1 \]which is in the vertex form \[ g(x) = a(x-h)^2 + k \].
02

- Determine the vertex

In the vertex form \( g(x) = a(x-h)^2 + k \), the vertex is given by the coordinates \((h, k)\). Here, \( h = -2 \) and \( k = -1 \). Therefore, the vertex is \((-2, -1)\).
03

- Find the axis of symmetry

The axis of symmetry of a quadratic function in vertex form is the vertical line \( x = h \). In this case, the axis of symmetry is \( x = -2 \).
04

- Determine the direction of the parabola

Since the coefficient \( a = \frac{3}{2} > 0 \), the parabola opens upwards. Therefore, the function has a minimum value at the vertex.
05

- Find the minimum value

The minimum value of the function occurs at the vertex. Since the y-coordinate of the vertex is \( -1 \), the minimum value is \( -1 \).
06

- Sketch the function

Plot the vertex \((-2, -1)\) and draw the parabola opening upwards. Use the axis of symmetry \( x = -2 \) as a reference to ensure the parabola is symmetric.

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Key Concepts

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

vertex form of a quadratic function
When dealing with quadratic functions, the vertex form is an incredibly useful way to understand their behavior. The vertex form of a quadratic function is \[ g(x) = a(x-h)^2 + k \] . In this formula:
  • \(a\) determines the direction and the width of the parabola.
  • \( (h, k) \) represents the vertex of the parabola.
Using the vertex form, you can quickly identify the vertex, which is the highest or lowest point of the parabola, depending on its direction.
axis of symmetry
The axis of symmetry is a vertical line that passes through the vertex of the parabola. For a quadratic function in the vertex form, \[ g(x) = a(x-h)^2 + k \],
  • The axis of symmetry has the equation \( x = h \).

This line divides the parabola into two mirror-image halves. For our function \ g(x)= \frac{3}{2}(x+2)^{2}-1 \, the axis of symmetry is
  • \( x = -2 \)
. Plotting this line helps visualize the parabola more accurately.
minimum and maximum values
Quadratic functions can either have a minimum or a maximum value, determined by the value at the vertex. Whether the parabola opens upwards or downwards will determine this:
  • If \( a > 0 \), the parabola opens upwards, and the function has a minimum value.
  • If \( a < 0 \), the parabola opens downwards, and the function has a maximum value.

In our case, since \( a = \frac{3}{2} > 0 \), the function
  • has a minimum value at the vertex \((-2, -1)\).
. Therefore, the minimum value of the function is
  • \(-1\).
parabola direction
The direction of the parabola is determined by the coefficient \(a\) in the quadratic function's vertex form: \[ g(x) = a(x-h)^2 + k \]
  • If \( a > 0 \), the parabola opens upwards.
  • If \( a < 0 \), the parabola opens downwards.
In our function \( g(x)= \frac{3}{2}(x+2)^{2}-1 \), we see that \( a = \frac{3}{2} \). Because \( \frac{3}{2} > 0 \),
  • the parabola opens upwards, indicating it has a minimum value at the vertex.
To sketch the graph, you plot the vertex and draw the parabola curving upwards from this point.

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