Problem 28 - Beginning and Intermediate Algebra

Question

Rewrite function in the form \(f(x)=a(x-h)^{2}+k\) by completing the square. Then, graph the function. Include the intercepts. \(g(x)=x^{2}+6 x+8\)

Step-by-Step Solution

Answer

We rewrite the given function \(g(x) = x^2 + 6x + 8\) by completing the square; we get \(g(x) = (x + 3)^2 - 1\), where \(a = 1, h = -3\), and \(k = -1\). The x-intercepts are \(-4\) and \(-2\), and the y-intercept is \((0, 8)\). To graph the function, plot the vertex \((-3, -1)\), the x-intercepts \((-4, 0)\) and \((-2, 0)\), and the y-intercept \((0, 8)\). The parabola opens upwards. Draw a smooth curve through these points.

Step 1: Rewrite the given function

First, let's rewrite the given function: g(x) = x^2 + 6x + 8

Step 2: Complete the square

Now, let's complete the square for the quadratic part (x^2 + 6x) of the given function. We need to find the value "c" such that when adding and subtracting it to the quadratic part, it forms a perfect square: (x^2 + 6x + c) - c c = (6/2)^2 = 9 Now the quadratic part becomes: (x^2 + 6x + 9) - 9

Step 3: Rewrite the function in vertex form

Now, we can rewrite g(x) in the vertex form, f(x) = a(x-h)^2 + k: g(x) = 1(x^2 + 6x + 9 - 9) + 8 g(x) = 1((x + 3)^2 - 9) + 8 g(x) = (x + 3)^2 - 1, where a = 1, h = -3, and k = -1.

Step 4: Find the intercepts

Now we need to find the x-intercepts (when g(x) = 0) and y-intercept (when x = 0). To find the x-intercepts, set g(x) to 0: 0 = (x + 3)^2 - 1 1 = (x + 3)^2 Now, take the square root of both sides: ±1 = x + 3 Subtract 3 to solve for x: x_intercepts: x = -4, -2 To find the y-intercept, set x to 0: g(0) = (0 + 3)^2 - 1 = 8 y_intercept: (0, 8)

Step 5: Graph the function

To graph g(x) = (x + 3)^2 - 1, follow these steps: 1. Plot the vertex (-3, -1). 2. Plot the x-intercepts (-4, 0) and (-2, 0). 3. Plot the y-intercept (0, 8). 4. Because "a" is positive, the parabola opens upwards. 5. Draw a smooth curve through the vertex, intercepts, and points symmetric to the vertex. Now, we have graphed the function g(x) = (x + 3)^2 - 1, with x-intercepts at (-4, 0) and (-2, 0) and a y-intercept at (0, 8).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function with a degree of two. This means that the highest exponent is 2. The general form of a quadratic function is expressed as:

• \(f(x) = ax^2 + bx + c\)

where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. The graph of a quadratic function is a curve called a parabola, which can either open upward or downward depending on the sign of \(a\).

If \(a\) is positive, the parabola opens upward, resembling a U-shape. But if \(a\) is negative, it opens downward, resembling an upside-down U-shape. Quadratic functions are found in many real-world scenarios such as projectile motion or optimal production problems.

Vertex Form

The vertex form of a quadratic function provides a simple way to graph a parabola by revealing its vertex directly. This form is represented as:

• \(f(x) = a(x-h)^2 + k\)

where \(a\), \(h\), and \(k\) are constants. Here, \((h, k)\) is the vertex of the parabola. The vertex is a crucial point since it's the peak or the lowest point of the parabola, depending on whether it opens upwards or downwards.

To convert a quadratic function from standard form \(f(x) = ax^2 + bx + c\) to vertex form, we use the method of completing the square. This involves rearranging the function to reveal a perfect square trinomial. Completing the square allows us to more easily identify the vertex and better understand the shape of the graph.

Intercepts

Intercepts are the points where the graph of a function crosses the axes. For a quadratic function, we frequently look for both the x-intercepts and the y-intercept.

• X-intercepts: These are the points where the graph touches or crosses the x-axis. To find the x-intercepts, we set \(f(x) = 0\) and solve for \(x\). For example, in the quadratic function \(g(x) = (x + 3)^2 - 1\), the x-intercepts are calculated as \(x = -4\) and \(x = -2\).
• Y-intercept: This is the point where the graph touches or crosses the y-axis (at \(x = 0\)). To find the y-intercept, substitute \(0\) for \(x\) in the function. For the function \(g(x) = (x + 3)^2 - 1\), the y-intercept occurs at \((0, 8)\).

Intercepts are important because they provide points that help define the shape and position of the parabola on the graph.

Graphing Parabolas

Graphing quadratic functions, or parabolas, involves plotting key points to define their shape and position on the coordinate plane. These points include the vertex, intercepts, and other interesting points revealed by symmetry.

Here’s a simple guide to graphing a parabola:

• Identify the vertex from the vertex form \((h, k)\) and plot it on the graph.
• Find and plot the x-intercepts, if they exist, by solving for when \(f(x) = 0\).
• Determine the y-intercept by calculating \(f(0)\) and plot it.
• Use symmetry around the vertex to plot additional points, ensuring you accurately reflect the curve's shape.
• Since \(a\) in the vertex form dictates the direction the parabola opens (upward for positive \(a\), downward for negative \(a\)), you can confirm the general shape before you complete the graph.

Once these points are plotted, draw a smooth and continuous curve through them to reveal the parabola's shape. This curve represents the path of the quadratic function across the graph.