Problem 17 - Beginning and Intermediate Algebra

Question

For quadratic function, identify the vertex, axis of symmetry, and \(x\)- and \(y\)-intercepts. Then, graph the function. \(f(x)=2(x-1)^{2}-8\)

Step-by-Step Solution

Answer

The vertex of the quadratic function \(f(x)=2(x-1)^{2}-8\) is (1, -8), the axis of symmetry is x = 1, the x-intercepts are -1 and 3, and the y-intercept is (0, -6). To graph the function, first plot the vertex, axis of symmetry, and intercepts, then sketch the parabolic curve that goes through those points.

Step 1: Find the vertex

The given function is in vertex form, which is: \(f(x)=a(x-h)^{2}+k\) Comparing the given function, \(f(x)=2(x-1)^{2}-8\), with the vertex form, we can identify the values of h and k that make up the vertex (h, k). In this case, h = 1 and k = -8. Therefore, the vertex of the function is (1, -8).

Step 2: Identify the axis of symmetry

The axis of symmetry is a vertical line that passes through the vertex. In this case, the axis of symmetry will be a vertical line passing through the x-coordinate of the vertex, which is x = 1. Therefore, the axis of symmetry is x = 1.

Step 3: Determine the x and y intercepts

To find the x-intercepts of the function, set the function equal to 0 and solve for x: \(0 = 2(x-1)^{2}-8\) Now, we need to isolate the x variable: \(8 = 2(x-1)^{2}\) \(\frac{8}{2} = (x-1)^{2}\) \(4 = (x-1)^{2}\) From here, we take the square root of both sides: ± \(\sqrt{4} = x-1\) Now, we can add 1: \(x = 1\pm\sqrt{4}\) We now have the x-intercepts: \(x = -1\) and \(x = 3\) To find the y-intercept, set x to 0 and solve for y: \(f(0) = 2(0-1)^{2}-8\) \(f(0) = 2(-1)^{2}-8\) \(f(0) = 2\cdot1-8\) \(f(0) = 2-8\) \(f(0) = -6\) Therefore, the y-intercept is (0, -6).

Step 4: Graph the function

Now that we have the vertex, axis of symmetry, and the x and y intercepts of the function, we can graph it. Plot the vertex (1, -8), the axis of symmetry (x = 1), the x-intercepts (-1, 0) and (3, 0), and the y-intercept (0, -6) on the graph. Then, sketch the quadratic function.

Key Concepts

Vertex

In a quadratic function written in the vertex form, which is \( f(x) = a(x-h)^2 + k \), the vertex plays a crucial role. It is the point \((h, k)\). This serves as the peak or the lowest point on the graph, depending on the orientation of the parabola.For the function \( f(x) = 2(x-1)^2 - 8 \), we identify the vertex by comparing it with the vertex form:

• \( h = 1 \)
• \( k = -8 \)

Thus, the vertex is \((1, -8)\). This indicates that the graph of the quadratic function has its minimum point at \((1, -8)\), since the parabola opens upwards due to the positive coefficient of \(a = 2\). The vertex is critical as it provides a clear point of reference for graphing and analyzes the parabolic structure.

Axis of Symmetry

The axis of symmetry for a quadratic function is a vertical line that divides the parabola into two congruent halves. This line always passes through the vertex and can be described by the equation \( x = h \), where \( h \) is the x-coordinate of the vertex. In the function \( f(x) = 2(x-1)^2 - 8 \), we have already identified the vertex at \((1, -8)\). Therefore:

• The axis of symmetry is \( x = 1 \).

This vertical line is crucial because it helps to visually balance the graph and ensure symmetry. When graphing, understanding the axis helps in plotting points accurately on both sides of the parabola.

Intercepts

Intercepts in a quadratic function are important points where the graph crosses the axes. This includes both x-intercepts and the y-intercept.

X-Intercepts

To find the x-intercepts, we set \( f(x) = 0 \) and solve for \( x \):

• Start with: \( 0 = 2(x-1)^2 - 8 \)
• Simplify to find \( x = -1 \) and \( x = 3 \).

These are the points \((-1, 0)\) and \((3, 0)\) where the parabola crosses the x-axis.

Y-Intercept

To find the y-intercept, we set \( x = 0 \):

• Calculate: \( f(0) = 2(0-1)^2 - 8 = -6 \)

The y-intercept is \((0, -6)\), which is where the parabola crosses the y-axis.These intercepts help in sketching the quadratic function as they provide key points that define the trajectory of the parabola on the graph.

Graphing a Quadratic Function

Graphing a quadratic function involves plotting its key components, such as the vertex, axis of symmetry, and intercepts, on a coordinate plane to visually represent the parabola.

Steps for Graphing

• Start by plotting the vertex \((1, -8)\). This is the lowest point of the parabola (since it opens upwards).
• Draw the axis of symmetry, a vertical line at \( x = 1 \), to aid in reflecting points evenly across the vertex.
• Plot the x-intercepts \((-1, 0)\) and \((3, 0)\), marking where the graph intersects the x-axis.
• Plot the y-intercept \((0, -6)\), where the graph crosses the y-axis.
• Using these points, sketch the parabola, ensuring the arms extend upwards on either side of the vertex.

By following these steps, you create a clear visual of the quadratic function \( f(x) = 2(x-1)^2 - 8 \) highlighting its symmetrical nature and curvature.