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Chapter 3: Problem 9

a) Find the vertex. b) Find the axis of symmetry. c) Determine whether there is a maximum or a minimum value and find that value. d) Graph the function. $$g(x)=\frac{x^{2}}{2}+4 x+6$$

Short Answer

Expert verified
Vertex: (-4, -2). Axis of symmetry: x = -4. Minimum value: -2.

Step by step solution

01

- Find the Vertex (h, k)

To find the vertex of the parabola represented by the function \( g(x) = \frac{x^2}{2} + 4x + 6 \), use the vertex formula \( h = -\frac{b}{2a} \). For this function, \( a = \frac{1}{2} \) and \( b = 4 \). Substitute these values into the formula: \[ h = -\frac{4}{2 \times \frac{1}{2}} = -\frac{4}{1} = -4 \] Next, find \( k \) by substituting \( h \) back into the function: \[ k = g(-4) = \frac{(-4)^2}{2} + 4(-4) + 6 = \frac{16}{2} - 16 + 6 = 8 - 16 + 6 = -2 \] So, the vertex is \( (-4, -2) \).
02

- Find the Axis of Symmetry

The axis of symmetry of a parabola given by \( g(x) = \frac{x^2}{2} + 4x + 6 \) has the equation \( x = h \). From the previous step, we know \( h = -4 \). Therefore, the axis of symmetry is \( x = -4 \).
03

- Determine Maximum or Minimum Value

The coefficient of the \( x^2 \) term (\( a = \frac{1}{2} \)) is positive, indicating that the parabola opens upwards and thus has a minimum value. From Step 1, the minimum value is the \( k \) coordinate of the vertex, which is \( k = -2 \).
04

- Graph the Function

To graph the function \( g(x) = \frac{x^2}{2} + 4x + 6 \): 1. Plot the vertex \( (-4, -2) \). 2. Draw the axis of symmetry, which is the vertical line \( x = -4 \). 3. Choose a few additional points on either side of the vertex and substitute their \( x \) values into the function to find corresponding \( y \) values. Plot these points. 4. Draw a smooth curve through all the points to complete the graph.

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Key Concepts

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

Vertex
The vertex is a key point in a quadratic function, represented as \( (h, k) \). For the given function \( g(x) = \frac{x^2}{2} + 4x + 6 \), the vertex formula \( h = -\frac{b}{2a} \) helps in finding the x-coordinate. Here, with \( a = \frac{1}{2} \) and \( b = 4 \), we calculate it as follows:
  • Substitute the values: \( h = -\frac{4}{2 \times \frac{1}{2}} = -\frac{4}{1} = -4 \)
Next, find the y-coordinate \( k \) by substituting \( h \) into the function:
  • Calculate: \( k = g(-4) = \frac{(-4)^2}{2} + 4(-4) + 6 = \frac{16}{2} - 16 + 6 = 8 - 16 + 6 = -2 \)
Hence, the vertex of the function is \( (-4, -2) \)
This point is the highest or lowest point on the graph depending on the parabola's direction. Instructionally, if \( a \) is positive, the vertex is a minimum point; if negative, a maximum point.
Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex of the parabola. This line divides the parabola into two symmetrical halves. For the function \( g(x) = \frac{x^2}{2} + 4x + 6 \), we previously found the vertex to be at \( x = -4 \). Thus, the equation for the axis of symmetry is:
  • \( x = -4 \)
This implies that on the graph, for every point \( (x, y) \) on one side of the axis, there is a corresponding point at \( (-4 - (x + 4), y) \) on the other side. This axis helps in graphing parabolas and understanding their symmetry.
Minimum Value
In a quadratic function \( ax^2 + bx + c \), if the coefficient \( a \) is positive, the parabola opens upwards, indicating that the function has a minimum value. For \( g(x) = \frac{x^2}{2} + 4x + 6 \), since \( a = \frac{1}{2} \), it is positive. Therefore, the parabola opens upwards and has a minimum value.

The minimum value occurs at the vertex, which we calculated to be at \( (-4, -2) \). This means the minimum value of the function is:
  • \( k = -2 \)
So, the function's minimum value is \( -2 \), occurring at \( x = -4 \). This point represents the lowest point on the graph.
Graphing Parabolas
To graph a parabola, start with the quadratic function \( g(x) = \frac{x^2}{2} + 4x + 6 \). Here's a step-by-step approach:
  • First, plot the vertex at \( (-4, -2) \).
  • Draw the axis of symmetry, the vertical line \( x = -4 \).
  • Select additional points on either side of the vertex. For instance, \( x = -3 \) and \( x = -5 \).
  • Substitute these x-values into the function to find corresponding y-values and plot these points. E.g., \( g(-3) = \frac{(-3)^2}{2} + 4(-3) + 6 = 1.5 - 12 + 6 = -4.5 \).
  • Plot more points if necessary for clarity.
Finally, draw a smooth curve connecting all the points. The resulting graph should be a U-shaped curve opening upwards, with the minimum value at \( (-4, -2) \). This technique transforms an equation into a visual representation, aiding understanding and problem-solving.

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