Problem 42 - 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. $$ f(x)=-0.25 x^{2}-3 x $$

Step-by-Step Solution

Answer

The y-intercept is (0, 0), the axis of symmetry is \( x = -6 \), and the vertex is (-6, 9). Graph using these points and additional values.

Step 1: Find the y-intercept

To find the y-intercept, set the value of \( x \) to 0 in the function. Substitute and solve: \[f(0) = -0.25(0)^2 - 3(0) = 0.\] So, the y-intercept is at the point (0, 0).

Step 2: Find the Axis of Symmetry

The axis of symmetry for a quadratic function in the form \( ax^2 + bx + c \) is given by the equation: \[x = -\frac{b}{2a}.\] Here, \( a = -0.25 \) and \( b = -3 \). Substitute these values: \[x = -\frac{-3}{2(-0.25)} = \frac{3}{-0.5} = -6.\] Therefore, the equation of the axis of symmetry is \( x = -6 \).

Step 3: Find the x-coordinate of the Vertex

The x-coordinate of the vertex is the same as the axis of symmetry: \( x = -6.\)

Step 4: Calculate the y-coordinate of the Vertex

Substitute \( x = -6 \) into the function to find the y-coordinate of the vertex: \[ f(-6) = -0.25(-6)^2 - 3(-6).\] Calculate \( f(-6) \): \[f(-6) = -0.25(36) + 18 = -9 + 18 = 9.\] So, the vertex of the parabola is at (-6, 9).

Step 5: Make a Table of Values

Choose values of \( x \) around the vertex \( x = -6 \). Include the vertex in the table: | \( x \) | \( f(x) \) | |-------|-------| | -8 | \( f(-8) = -0.25(64) + 24 = 8 \) | | -7 | \( f(-7) = -0.25(49) + 21 = 8.75 \) | | -6 | 9 | | -5 | \( f(-5) = -0.25(25) + 15 = 8.75 \) | | -4 | \( f(-4) = -0.25(16) + 12 = 8 \) |

Step 6: Graph the Function

Using the y-intercept, axis of symmetry, vertex, and table of values, plot the points on a graph: - Mark the y-intercept at (0, 0).- Draw the axis of symmetry at \( x = -6 \).- Plot the vertex at (-6, 9).- Plot the other points from the table.- Draw a smooth curve through these points to create the parabola.

Key Concepts

Understanding the Y-Intercept

The y-intercept of a quadratic function is the point where the graph crosses the y-axis. This happens when the value of \( x \) is zero. For the given function \( f(x) = -0.25x^2 - 3x \), you find the y-intercept by substituting \( x = 0 \) into the equation. This gives you

• \( f(0) = -0.25(0)^2 - 3(0) = 0 \)

Therefore, the y-intercept is at the origin, point (0, 0). This is a crucial point because it gives you the initial position of the parabola along the y-axis. Whenever you find the y-intercept, you're essentially identifying one of the essential starting features in graphing quadratic functions.

Finding the Axis of Symmetry

The axis of symmetry in a quadratic function describes a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex of the parabola, and its equation is derived from the quadratic formula in standard form \( ax^2 + bx + c \). The formula to find the axis of symmetry is:

• \( x = -\frac{b}{2a} \)

In the quadratic function \( f(x) = -0.25x^2 - 3x \), you have \( a = -0.25 \) and \( b = -3 \). Substitute these values into the equation:

• \( x = -\frac{-3}{2(-0.25)} = -6 \)

This means your axis of symmetry is the vertical line \( x = -6 \). Understanding this line helps in graphing because it guides where other points will mirror across the parabola.

Vertex Calculation and Its Importance

The vertex of a quadratic function is an essential feature, representing the highest or lowest point on the graph, depending on the parabola's direction. This point lies on the axis of symmetry. Since we found the axis of symmetry to be \( x = -6 \), this is also the \( x \)-coordinate of the vertex.Next, to find the y-coordinate, substitute \( x = -6 \) into the function \( f(x) = -0.25x^2 - 3x \):

• \( f(-6) = -0.25(-6)^2 - 3(-6) \)
• \( f(-6) = -9 + 18 = 9 \)

Therefore, the vertex is at the point \((-6, 9)\). This point helps you understand the direction and extremity of the parabola, making it pivotal for graphing.

Graphing Quadratic Functions

Graphing quadratic functions involves plotting points and understanding the structure of a parabola. You start with the y-intercept and the vertex, which provide key points of reference. For the function \( f(x) = -0.25x^2 - 3x \), the y-intercept is at (0, 0) and the vertex at \((-6, 9)\).To graph accurately, create a table of values around the vertex. You'll choose \( x \)-values both left and right of the vertex:

• \( x = -8 \), \( f(-8) = 8 \)
• \( x = -7 \), \( f(-7) = 8.75 \)
• \( x = -6 \), \( f(-6) = 9 \)
• \( x = -5 \), \( f(-5) = 8.75 \)
• \( x = -4 \), \( f(-4) = 8 \)

Plot these points on a graph, draw the axis of symmetry at \( x = -6 \), and smoothly connect the points to form a parabolic curve. The symmetry and shape you get will help you visualize the function effectively. The parabola's opening direction and curvature come from the sign and value of \( a \) in the function, indicating how wide or narrow the curve will be. Always remember, the parabola will be symmetric along the axis you've determined.