Problem 39 - 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. $$ f(x)=-3 x^{2}-4 x $$

Step-by-Step Solution

Answer

\( y \)-intercept: \((0, 0)\), axis of symmetry: \(x=\frac{2}{3}\), vertex: \(\left(\frac{2}{3}, -4 \right)\).

Step 1: Identify the Quadratic Function Components

The given quadratic function is \( f(x) = -3x^2 - 4x \). This can be compared to the standard form \( ax^2 + bx + c \), where \( a = -3 \), \( b = -4 \), and \( c = 0 \).

Step 2: Find the Y-Intercept

The \( y \)-intercept is the value of the function when \( x = 0 \). Substitute \( x = 0 \) into \( f(x) \): \( f(0) = -3(0)^2 - 4(0) = 0 \). Thus, the \( y \)-intercept is \( (0, 0) \).

Step 3: Find the Axis of Symmetry

The axis of symmetry for a quadratic function \( ax^2 + bx + c \) is found using the formula \( x = -\frac{b}{2a} \). Here, \( b = -4 \) and \( a = -3 \), so \( x = -\frac{-4}{2(-3)} = \frac{4}{6} = \frac{2}{3} \). Thus, the axis of symmetry is \( x = \frac{2}{3} \).

Step 4: Find the X-Coordinate of the Vertex

The \( x \)-coordinate of the vertex is the same as the axis of symmetry, \( x = \frac{2}{3} \).

Step 5: Calculate the Y-Coordinate of the Vertex

To find the \( y \)-coordinate of the vertex, substitute \( x = \frac{2}{3} \) back into the function: \( f\left(\frac{2}{3}\right) = -3\left(\frac{2}{3}\right)^2 - 4\left(\frac{2}{3}\right) = -3 \times \frac{4}{9} - \frac{8}{3} = -\frac{4}{3} - \frac{8}{3} = -\frac{12}{3} = -4 \). Hence, the vertex is \( \left( \frac{2}{3}, -4 \right) \).

Step 6: Construct a Table of Values

Select values for \( x \) around the vertex and calculate the corresponding \( y \)-values. For example: - \( x = 0 \): \( f(0) = 0 \) - \( x = 1 \): \( f(1) = -7 \) - \( x = \frac{2}{3} \): \( f\left(\frac{2}{3}\right) = -4 \) - \( x = -1 \): \( f(-1) = -1 \) - \( x = -2 \): \( f(-2) = 4 \). This table of values will help for graphing.

Step 7: Graph the Function

Using the table of values, plot each point on a coordinate plane: the points are \((0, 0)\), \((1, -7)\), \(\left(\frac{2}{3}, -4\right)\), \((-1, -1)\), and \((-2, 4)\). Draw a smooth parabolic curve through these points.

Key Concepts

Y-Intercept

In the world of quadratic functions, the y-intercept is where the graph of the function meets the y-axis. This point is crucial because it gives one direct piece of the puzzle when plotting the graph. To determine the y-intercept, you simply set the value of \(x\) to zero in the function. By doing this, any term with \(x\) drops off, leaving you with the constant term \(c\).For the function given in this exercise, \(f(x) = -3x^2 - 4x\), the absence of a constant term (\(c = 0\)) means when \(x = 0\):

• \(f(0) = -3(0)^2 - 4(0) = 0\)

So, the y-intercept is the point \((0, 0)\). This tells us that the graph crosses the y-axis right at the origin.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. This is a crucial feature of quadratic functions because it helps pinpoint the location of the vertex, giving symmetry to the parabolic shape.You can find the axis of symmetry using the formula:

• \(x = -\frac{b}{2a}\)

Here, for our quadratic function \(f(x) = -3x^2 - 4x\), we identify:

• \(a = -3\)
• \(b = -4\)

Plug these into the formula:

• \(x = -\frac{-4}{2(-3)} = \frac{4}{-6} = \frac{2}{3}\)

So, the axis of symmetry is the line \(x = \frac{2}{3}\). This line not only helps in graphing but also aids in understanding the parabola's balance about this line.

Vertex

The vertex of a quadratic function is a significant point where the curvature turns direction. It can either be the highest or lowest point on the graph, depending on the parabola opening direction.With the axis of symmetry known to be \(x = \frac{2}{3}\), this is also the \(x\)-coordinate for the vertex. To completely determine the vertex, we need to find the \(y\)-coordinate by substituting \(x = \frac{2}{3}\) back into the original function:

• \(f\left(\frac{2}{3}\right) = -3 \times \left(\frac{2}{3}\right)^2 - 4 \times \left(\frac{2}{3}\right)\)
• \(-3 \times \frac{4}{9} - \frac{8}{3} = -\frac{4}{3} - \frac{8}{3} \)
• = \(-\frac{12}{3} = -4\)

Thus, the vertex is at \(\left(\frac{2}{3}, -4\right)\). Understanding the vertex's location is key to sketching the parabola accurately and gives insight into its maximum or minimum value.