Problem 36 - 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. \(g(x)=-x^{2}-4 x-6\)

Step-by-Step Solution

Answer

The function \(g(x)=-x^{2}-4x-6\) can be rewritten in vertex form by completing the square. This results in the function \(g(x)=-1(x+2)^2+2\) with a vertex at (-2, 2), a y-intercept at (0, -2), and x-intercepts at \((-2+\sqrt{2}, 0)\) and \((-2-\sqrt{2}, 0)\). The graph of the function is a parabola opening downward.

Step 1: Rewrite the function in vertex form

To rewrite the function in the form \(f(x)=a(x-h)^{2}+k\), we need to complete the square: \(g(x)=-x^{2}-4x-6\) a = -1, since the coefficient of \(x^{2}\) is -1. First, factor out the "a" value from the quadratic and linear terms: \(g(x)=-1(x^{2}+4x)-6\) Now, we will complete the square inside the parenthesis. To do this, we need to find the value "c" that allows us to write \(x^2+4x+c\) as a perfect square. c is calculated using the formula \(\frac{b}{2}^2\), where b is the coefficient of the linear term: \(c =\frac{4}{2}^{2} = 4\) Now, add and subtract the value of "c" inside the parentheses, keeping in mind the "a" value outside the parentheses: \(g(x)=-1(x^{2}+4x+4-4)-6\) Simplify by combining the -4 and the -6 outside the parentheses: \(g(x)=-1(x^{2}+4x+4)+2\) Now, rewrite the quadratic inside parentheses as a perfect square: \(g(x) = -1(x+2)^{2}+2\) The rewritten function is in the form \(f(x)=a(x-h)^{2}+k\), with \(a=-1\), \(h=-2\), and \(k=2\).

Step 2: Graph the function

To graph the function, we first need to identify important points: 1. The vertex: This is given by (h, k) = (-2, 2). 2. The y-intercept: To find the y-intercept, we set x to 0 and solve for y: \(g(0) = -1(0+2)^{2}+2 = -1(2^2)+2 = -4+2 = -2\) This means the y-intercept is at (0, -2). 3. The x-intercepts: To find the x-intercepts, we set y to 0 and solve for x: \(0 = -1(x+2)^{2}+2\) First, subtract 2 from both sides: \(-2 = -1(x+2)^{2}\) Now, we divide both sides by -1: \(2=(x+2)^{2}\) Take the square root of both sides: \(\sqrt{2}=x+2\) or \(-\sqrt{2}=x+2\) Solve for x: \(x = -2+\sqrt{2}\) or \(x = -2-\sqrt{2}\) Now, we have the vertex, y-intercept, and x-intercepts, all the necessary points, to graph the function: 1. Plot the vertex at (-2, 2). 2. Plot the y-intercept at (0, -2). 3. Plot the x-intercepts at \((-2+\sqrt{2}, 0)\) and \((-2-\sqrt{2}, 0)\). 4. Draw the parabola with these points, and notices that it opens downward since \(a=-1\). The graph is now complete.

Key Concepts

Vertex Form

The vertex form of a quadratic function is a powerful way to understand the shape and position of a parabola. It is expressed as \(f(x) = a(x-h)^2 + k\), where \(a\), \(h\), and \(k\) are constants. Here, the role of these constants is critical:

• \(a\) determines the direction of the parabola (upward if positive, downward if negative) and its width.
• \(h\) and \(k\) represent the coordinates of the vertex, which is the "turning point" of the parabola.

Translating a standard quadratic form to vertex form involves a technique called "completing the square." This process helps in visualizing the function as a transformation of \(x^2\), shifting it horizontally by \(h\) and vertically by \(k\). Let's say we have a quadratic in standard form \(g(x) = -x^2 - 4x - 6\). By completing the square, we find it in vertex form: \(-1(x + 2)^2 + 2\), revealing a vertex at \((-2, 2)\).
The vertex form thus provides a direct way to read important graph features, such as the highest or lowest point of the graph, simply by looking at \(h\) and \(k\). This makes understanding and sketching the graph significantly easier.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, taking the form \(f(x) = ax^2 + bx + c\). These functions create a curve known as a parabola when graphed. Key characteristics include:

• The parabola's symmetry around a vertical line called the axis of symmetry, found at \(x = -\frac{b}{2a}\).
• The vertex, which is the maximum or minimum point of the function, determined from the vertex form as \((h, k)\).
• Determination of the parabola’s opening direction and width by the leading coefficient \(a\).

For our function \(g(x) = -x^2 - 4x - 6\), the quadratic nature means it will form a parabola. Since \(a = -1\) here, the parabola opens downwards, indicating its vertex is the maximum point. By converting to vertex form or using the formula to find the vertex, you can easily determine it at \((-2, 2)\). This form also aids in finding points like the y-intercept (where \(x=0\)) and x-intercepts (where the graph crosses the x-axis).
Quadratic functions elegantly model real-world scenarios with such curved paths, from projectile motion to optimizing areas. Understanding this can immensely help in both academic and practical life applications.

Parabola Graphing

Graphing parabolas involves plotting its key points and understanding its direction of opening. Start by locating the vertex, as this marks a central point in your graph. For \(g(x) = -1(x + 2)^2 + 2\), the vertex is \((-2, 2)\). This point allows you to draw the axis of symmetry, a line where one side of the parabola reflects onto the other.
Next, find the y-intercept by substituting \(x=0\) into the function, resulting in \((0, -2)\) for this function. X-intercepts are found by setting \(y=0\) and solving the equation, giving us \((-2 + \sqrt{2}, 0)\) and \((-2 - \sqrt{2}, 0)\).
Once these points are plotted, drawing the curve is the final step. Remember that in this exercise, since \(a = -1\), the parabola opens downward. This means it starts at the vertex \((-2, 2)\) and extends downwards towards the x-axis. The graph should be a smooth curve passing through these intercepts and symmetrical about its axis.
Graphing helps visualize how changes in quadratic equations affect their corresponding parabolas. This understanding is crucial for topics like solving quadratic equations and analyzing real-world data.