Problem 24 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=-2(x+3)^{2}-4$$

Step-by-Step Solution

Answer

Plot the vertex \((-3, -4)\) and additional points like \((-2, -6)\) and \((-4, -6)\) to sketch the downward opening parabola.

Step 1: Identify the Function Type

The given function is in the form of a quadratic function, specifically in vertex form, which is \(f(x) = a(x-h)^2 + k\). Here, \(a = -2\), \(h = -3\), and \(k = -4\). The vertex form clearly shows that this is a parabola.

Step 2: Determine the Vertex

The vertex of a parabolic function in vertex form \(f(x) = a(x-h)^2 + k\) is given by the point \((h, k)\). For this function, \(h = -3\) and \(k = -4\), so the vertex is at the point \((-3, -4)\).

Step 3: Determine the Direction of the Parabola

The coefficient \(a = -2\) is negative, which means the parabola opens downward. The negative sign indicates that the parabola is reflected over the x-axis.

Step 4: Find Additional Points

To plot the graph, calculate additional points. Choose values for \(x\) around the vertex and calculate \(f(x)\). For example, for \(x = -2\), \(f(-2) = -2((-2)+3)^2 -4 = -2(1)^2 - 4 = -2 - 4 = -6\). Similarly, calculate for \(x = -4\) as another example: \(f(-4) = -2((-4)+3)^2 -4 = -2(-1)^2 - 4 = -2 - 4 = -6\).

Step 5: Plot the Points and Sketch the Graph

Plot the vertex \((-3, -4)\) and the calculated points \((-2, -6)\) and \((-4, -6)\) on a coordinate plane. Since the parabola is symmetrical about the line \(x = -3\), draw a smooth curve through these points, ensuring the parabola opens downward.

Key Concepts

Vertex Form of a Quadratic Equation

The vertex form of a quadratic equation is particularly useful for easily identifying the key features of a parabola. It is expressed as:\[ f(x) = a(x-h)^2 + k \]Here, \(a\), \(h\), and \(k\) are constants with specific roles:

• \(a\) determines the “width” and direction of the parabola (how “wide” or “narrow” it looks).
• \((h, k)\) represents the vertex of the parabola, the highest or lowest point depending on the direction.

When using the vertex form, you can quickly locate the vertex. In our example, \(f(x) = -2(x+3)^2 - 4\), it shows us that the vertex is \((-3, -4)\). This form is preferred when graphing because it allows easy plotting of the vertex directly, giving a head start in drawing the parabola.

Parabola Properties

Understanding the properties of a parabola is crucial when interpreting or graphing quadratic functions. These properties include:

• Vertex: This is the turning point of the parabola. As we calculated, for \(f(x) = -2(x+3)^2 - 4\), our vertex is at \((-3, -4)\).
• Direction: The sign of \(a\) determines whether the parabola opens upwards or downwards. Here, \(a = -2\), and because it is negative, the parabola opens downwards.
• Axis of Symmetry: The parabola is symmetric about the line \(x = h\), making the axis of symmetry \(x = -3\) in this case.
• Width: Value of \(a\) affects how "wide" the parabola appears. A smaller absolute value of \(a\) results in a wider shape, while a larger value makes it narrower. Our \(a = -2\) implies a steeper parabola compared to \(a = -1\).

These properties help both in graphing and understanding the behaviour of quadratic functions.

Plotting Graphs

Plotting graphs of quadratic functions involves a few straightforward steps. Here's how you can plot the function \(f(x) = -2(x+3)^2 - 4\):Begin by plotting the vertex. In our function, the vertex is at \((-3, -4)\) so place this point first on a coordinate plane. Next, calculate additional points by choosing \(x\)-values around \(h\). For example, using \(x = -2\) and \(x = -4\), we calculated points \((-2, -6)\) and \((-4, -6)\), respectively.

• Plot these additional points for accuracy.
• Draw the axis of symmetry through \(x = -3\) and ensure symmetry in your plot.
• Connect these points smoothly to form the parabola, ensuring the curve opens downward, as determined by the negative \(a\).

This method of plotting ensures clarity and correctness in graphing quadratic functions.