Problem 41 - Algebra 2

Question

Complete parts a-c for each quadratic function. a. Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x\) -coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. $$ f(x)=0.5 x^{2}-1 $$

Step-by-Step Solution

Answer

Y-intercept: (0, -1); Axis of Symmetry: \(x = 0\); Vertex: (0, -1); Graph is symmetric about y-axis.

Step 1: Finding the Y-Intercept

To find the y-intercept of the function, substitute \( x = 0 \) into the function \( f(x) = 0.5x^2 - 1 \). This gives \( f(0) = 0.5(0)^2 - 1 = -1 \). The y-intercept is \( (0, -1) \).

Step 2: Identifying the Axis of Symmetry

The equation of the axis of symmetry for a quadratic function of the form \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \). Here, \( a = 0.5 \) and \( b = 0 \), so the axis of symmetry is \( x = -\frac{0}{2(0.5)} = 0 \).

Step 3: Determining the Vertex

The vertex of a quadratic function is on the axis of symmetry. Since the axis is \( x = 0 \), the x-coordinate of the vertex is 0. Substitute \( x = 0 \) back into the function to find the y-coordinate: \( f(0) = -1 \). Therefore, the vertex is \( (0, -1) \).

Step 4: Creating a Table of Values

Choose a few values of \( x \) around the vertex to create a table. For example, \[ x: -2, -1, 0, 1, 2 \] and calculate \( f(x) \): \(-2: 1, -1: -0.5, 0: -1, 1: -0.5, 2: 1\). The table is: \((-2, 1), (-1, -0.5), (0, -1), (1, -0.5), (2, 1)\).

Step 5: Graphing the Function

Using the points from the table and the vertex, plot them on a graph. From these points, you will see the shape of the parabola. The y-intercept and vertex \((0, -1)\) helps in placing the graph accurately. Draw a smooth curve through these points making sure to respect the symmetry about the vertical line \( x = 0 \).

Key Concepts

Y-Intercept

The y-intercept of a quadratic function is the point where the graph intersects the y-axis. To find it, we set the value of \( x \) in the equation to 0. In our case, the function is given as \( f(x) = 0.5x^2 - 1 \). By substituting \( x = 0 \), we get:

• \( f(0) = 0.5(0)^2 - 1 = -1 \)

Thus, the y-intercept is the point \((0, -1)\). This means the parabola crosses the y-axis at \( y = -1 \). Locating the y-intercept is crucial for graphing since it sets the initial position of the graph on the coordinate plane. Keep this point in mind as you plot additional points or follow the graph's direction.

Axis of Symmetry

The axis of symmetry is a vital component of any quadratic function. It is a vertical line that divides the parabola into two mirror-image halves. For a standard quadratic equation \( ax^2 + bx + c \), the axis of symmetry can be calculated using the formula \( x = -\frac{b}{2a} \). Let's apply this formula:

• For this function, \( a = 0.5 \) and \( b = 0 \).
• The axis of symmetry is \( x = -\frac{0}{2 \times 0.5} = 0 \).

In this particular function, the axis of symmetry is along the line \( x = 0 \). This line will be crucial when sketching the parabola because it shows where the function will "fold" symmetrically.

Vertex

The vertex of a parabola is the point where it changes direction. It is either the maximum or minimum point of the curve, depending on the orientation. For the quadratic function \( f(x) = 0.5x^2 - 1 \), the vertex lies on the axis of symmetry.Given the axis of symmetry \( x = 0 \), we substitute \( x \) back into the function to find the y-coordinate:

• \( f(0) = 0.5(0)^2 - 1 = -1 \)

Thus, the vertex of this parabola is at \((0, -1)\). Understanding the vertex allows us to guide the direction of the parabola and serves as a reference point for graphing.

Table of Values

Creating a table of values is a straightforward method of identifying additional points to graph the parabola. This further solidifies our understanding and assists in accurately sketching the graph. When selecting values, it is smart to choose points around the vertex:

• Choose points: \( x = -2, -1, 0, 1, 2 \)

Calculate \( f(x) \) for each:

• \( (-2, 1) \)
• \( (-1, -0.5) \)
• \( (0, -1) \)
• \( (1, -0.5) \)
• \( (2, 1) \)

These points give us a structured approach to graphing, confirming the shape of the parabola created from these calculations.

Graphing Parabolas

Graphing a quadratic function involves plotting the y-intercept, vertex, and additional points derived from the table of values. With these points, we can visualize the parabola's shape. Ensure you note the axis of symmetry to maintain balance in the graph:1. Start by plotting the vertex \((0, -1)\). This is your guide for direction.2. Plot the y-intercept \((0, -1)\); it coincides with the vertex.3. Use the calculated points from the table: \((-2, 1), (-1, -0.5), (1, -0.5), (2, 1)\).4. Draw the parabola by linking the points with a smooth curve.5. Make sure the curve is symmetric with respect to the axis of symmetry \( x = 0 \).By following these steps, you'll accurately graph the quadratic function, showing a clear representation of the relationship between \( x \) and \( f(x) \).