Problem 36 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=(x+5)^{2}-2$$

Step-by-Step Solution

Answer

The graph is a parabola with vertex at (-5, -2) that opens upward.

Step 1: Identify the Function Type

The function given is \( f(x) = (x+5)^2 - 2 \). This is a quadratic function written in vertex form, \( f(x) = a(x-h)^2 + k \), where the vertex \( (h,k) \) is a key point of the graph.

Step 2: Find the Vertex of the Parabola

To find the vertex of the parabola, identify \( h \) and \( k \) from the function \( (x+5)^2 - 2 \). Here, \( h = -5 \) and \( k = -2 \), so the vertex is \( (-5, -2) \).

Step 3: Determine the Axis of Symmetry

The axis of symmetry of a parabola in vertex form \( a(x-h)^2 + k \) is \( x = h \). For \( f(x) = (x+5)^2 - 2 \), the axis of symmetry is \( x = -5 \).

Step 4: Identify the Direction of the Parabola

The coefficient \( a \) in \( a(x-h)^2 + k \) determines the opening direction of the parabola. Since \( a = 1 \) is positive, the parabola opens upward.

Step 5: Plot Additional Points

Choose additional x-values to determine more points on the graph. For example, calculate \( f(-4) = (-4+5)^2 - 2 = 1^2 - 2 = -1 \), thus the point \((-4, -1)\). Similarly, calculate \( f(-6) = (-6+5)^2 - 2 = (-1)^2 - 2 = -1 \), giving the point \((-6, -1)\).

Step 6: Draw the Parabola

Using the vertex \((-5, -2)\), the axis of symmetry \(x = -5\), and the additional points, sketch the parabola. The axis of symmetry helps ensure the parabola is symmetrical as it opens upwards.

Key Concepts

Vertex Form

A quadratic function in vertex form gives us a simple way to identify the key characteristics of a parabola's graph. It is expressed as \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.
The "vertex" of a parabola is its highest or lowest point, depending on the direction it opens. The vertex form is particularly useful because it tells us the vertex with just a glance at the equation. In this form:

• \( h \) represents a horizontal shift from the origin, moving the parabola left or right.
• \( k \) is the vertical shift, moving it up or down.

For example, in the equation \( f(x) = (x+5)^2 - 2 \), the vertex is located at \((-5, -2)\). This is determined directly by observing that \( h = -5 \) and \( k = -2 \). Knowing the vertex is crucial because it is the central point from which the parabola is plotted.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, effectively dividing it into two mirror-image halves. For any quadratic function in vertex form \( a(x-h)^2 + k \), the axis of symmetry is always given by \( x = h \). This is because the vertex is the most left-right balanced point on the parabola.
Therefore, for the function \( f(x) = (x+5)^2 - 2 \), the axis of symmetry is \( x = -5 \). This tells us that if you were to "fold" the graph along this line, both sides would match perfectly.
The axis of symmetry is a powerful tool in graphing parabolas because it helps in planning the layout of the graph. By plotting key points both right and left of the axis, we maintain symmetry, ensuring that the parabola is drawn correctly.
Remember, even if the parabola opens upwards or downwards, the axis of symmetry will always pass through the vertex.

Parabola Graphing

Graphing a parabola may seem tricky, but with a few simple steps, it becomes easier. First, ensuring the equation is in vertex form makes it straightforward to identify the vertex, which is our starting point.
To graph \( f(x) = (x+5)^2-2 \):

• Start by plotting the vertex \((-5, -2)\) on the coordinate plane.
• Next, draw the axis of symmetry, which is \( x = -5 \) in this case.
• Identify whether the parabola opens upwards or downwards by examining the coefficient \( a \). Since \( a = 1 \) is positive, the parabola opens upwards.

With the vertex and direction established, plot additional points on either side of the axis. For example, if you calculate \( f(-4) = -1 \) and \( f(-6) = -1 \), add these points to the graph. These points confirm the curvature and help complete the parabola.
Using both the vertex and axis of symmetry as guides, sketch the parabola, ensuring it is symmetrical and accurately reflects all the calculated points.
Understanding each of these elements simplifies the process of graphing and ensures accuracy. It's a strategic approach that empowers you to draw precise and consistent parabolas on the graph.