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Chapter 8: Problem 36

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

Short Answer

Expert verified
Vertex: (-3, -2). Axis of symmetry: x = -3. Minimum value: -2. The parabola opens upwards.

Step by step solution

01

Identify the function's form

The given function is \[ f(x) = (x+3)^2 -2 \] which is a quadratic function in the form \[ f(x) = a(x-h)^2 + k \] where \( a = 1 \), \( h = -3 \), and \( k = -2 \).
02

Find the vertex

The vertex \((h, k)\) is given by the values of \( h \) and \( k \) in the vertex form. Therefore, the vertex is \[ (-3, -2) \].
03

Determine the axis of symmetry

The axis of symmetry for a parabola in the form \[ f(x) = a(x-h)^2 + k \] is the vertical line \( x = h \). Therefore, for the given function, the axis of symmetry is \[ x = -3 \].
04

Identify the direction of the parabola

Since the coefficient \( a = 1 \) is positive, the parabola opens upwards. This means the function has a minimum value.
05

Find the minimum value

The minimum value of the function \[ f(x) = a(x-h)^2 + k \] occurs at the vertex. Therefore, the minimum value is \[ f(-3) = -2 \].
06

Graph the function

To graph the function, plot the vertex \((-3, -2)\). Then draw the parabola opening upwards, making sure it is symmetrical around the axis of symmetry \( x = -3 \).

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Key Concepts

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

Vertex Form of a Parabola
The vertex form of a quadratic function is a key concept in graphing and analyzing parabolas. It is written as follows: \text { f(x) = a(x-h)^2 + k } Here,
  • \(a\): Determines the width and direction (upward if positive, downward if negative) of the parabola.
  • \(h\): The \(x\)-coordinate of the vertex.
  • \(k\): The \(y\)-coordinate of the vertex.
For the function \(f(x) = (x+3)^2 - 2\), we can identify the values as \(a = 1\), \(h = -3\), and \(k = -2\). This tells us that the vertex of the parabola is at \((-3, -2)\), making it easier to plot and understand the function鈥檚 behavior. Knowing these values simplifies finding the vertex and other related properties of the parabola.
Axis of Symmetry
The axis of symmetry in a quadratic function is a vertical line that passes through the vertex. This line divides the parabola into two mirrored halves. For the vertex form \text{ f(x) = a(x-h)^2 + k }, the axis of symmetry is given by the equation \text{ x = h }. In our example function \(f(x) = (x+3)^2 - 2\), we identified \(h = -3\). Therefore, the axis of symmetry is \text{ x = -3 }.This line is crucial when graphing because it shows where the parabola turns. Every point on one side of the axis has a corresponding point on the other side. Understanding the axis helps in accurately drawing the shape of the parabola.
Minimum Value of a Quadratic Function
For quadratic functions that open upwards (\(a > 0\)), the vertex represents the minimum point of the parabola. The minimum value is simply the \(y\)-coordinate of the vertex. The general form of our quadratic function is \text{ f(x) = a(x-h)^2 + k }.The minimum value is \text{ k }In the function \(f(x) = (x+3)^2 - 2\), we already determined that \(k = -2\). Thus, the minimum value of the function is \(-2\). Knowing where the minimum value is helps in understanding the range of the quadratic function and constraints in practical problems.
Graphing Parabolas
Graphing a parabola involves several steps, starting with identifying key features from the vertex form of the function. These steps include:
  • Identify the vertex \((h, k)\). For \(f(x) = (x+3)^2 - 2\), it鈥檚 \((-3, -2)\).
  • Draw the axis of symmetry, which is the line \(x = h\). Here, \(x = -3\).
  • Plot the vertex on the coordinate plane.
  • Determine the direction of the parabola. Since \(a > 0\), it opens upwards.
  • Identify additional points on either side of the axis by choosing \(x eq h\) values.
  • Plot these points and draw the parabola shape, ensuring symmetry.
By following these steps, you can accurately sketch the quadratic function, helping you visualize its behavior and interpret its properties in various contexts.

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