Problem 31 - A Graphical Approach to Precalculus with Limits

Question

For quadratic function, (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=-3 x^{2}+24 x-46$$

Step-by-Step Solution

Answer

The vertex is (4, 2) and the parabola opens downward.

Step 1: Identify the coefficients

For the quadratic function \( P(x) = -3x^2 + 24x - 46 \), identify the coefficients: \( a = -3 \), \( b = 24 \), and \( c = -46 \). These will be used to find the vertex.

Step 2: Find the x-coordinate of the vertex

Use the vertex formula for the x-coordinate: \( x = \frac{-b}{2a} \). Substitute \( b = 24 \) and \( a = -3 \) into this formula: \( x = \frac{-24}{2(-3)} = \frac{-24}{-6} = 4 \).

Step 3: Calculate the y-coordinate of the vertex

Substitute \( x = 4 \) into the function to find \( P(4) \): \( P(4) = -3(4)^2 + 24(4) - 46 \). Calculate: \( P(4) = -3(16) + 96 - 46 = -48 + 96 - 46 = 2 \). So the y-coordinate is 2.

Step 4: Vertex coordinates

The vertex of the quadratic function is \((4, 2)\) after substituting into the formula for the vertex and calculating the y-coordinate.

Step 5: Graph the quadratic function

To graph \( P(x) = -3x^2 + 24x - 46 \), start with the vertex at \((4, 2)\). Since \( a = -3 \) is negative, the parabola opens downwards. Plot additional points to define the shape by choosing values of \( x \) around the vertex and calculate their corresponding \( P(x) \) values, then draw a smooth curve through these points.

Key Concepts

Vertex Formula

The vertex formula is a key tool when dealing with quadratic functions. It helps you find the vertex of a parabola, which is the highest or lowest point on its graph, depending on the orientation of the parabola. The formula is written as:

• For the x-coordinate, use: \( x = \frac{-b}{2a} \)
• To find the y-coordinate, substitute the x value back into the function.

This method is valuable because once you know the vertex, you understand a lot about the graph's shape. In our equation \( P(x) = -3x^2 + 24x - 46 \), we identified the coefficients \( a = -3 \) and \( b = 24 \) using the vertex formula:

• Substitute to find \( x = \frac{-24}{2(-3)} = 4 \)
• Substitute \( x = 4 \) back to find \( y \) and get \( P(4) = 2 \)

This shows the vertex of the parabola is at \((4, 2)\). Having this information allows you to proceed with graphing the quadratic function.

Graphing Quadratic Functions

Graphing quadratic functions is an important skill in math that visually represents the solutions and characteristics of a quadratic equation. The process starts by locating the vertex as calculated using the vertex formula.
For the function \( P(x) = -3x^2 + 24x - 46 \), the vertex was found at \((4, 2)\). This is your starting point for graphing.
Since the coefficient \( a = -3 \) is negative, the parabola opens downward. Plotting points around the vertex helps define the curve. Choose some \( x \) values near \( 4 \), calculate their corresponding \( P(x) \) values, and plot these points:

• For example, if \( x = 3 \), calculate \( P(3) \)
• Similarly, calculate \( P(5) \) based on the function

Connect these points smoothly to form a downward-opening parabola. Remember, the vertex is the peak of the parabola in this case.

Coefficients of Quadratic Equations

Understanding the coefficients of a quadratic equation is fundamental, as they dictate the shape and position of the parabola on a graph. A general quadratic equation is written in the form \( ax^2 + bx + c \), where:

• \( a \) is the leading coefficient
• \( b \) is the linear coefficient
• \( c \) is the constant term

For the equation \( P(x) = -3x^2 + 24x - 46 \):

• \( a = -3 \), which affects the direction and width of the parabola. Since \( a \) is negative, the parabola opens downwards, and a larger absolute value of \( a \) means a narrower parabola.
• \( b = 24 \) influences the vertex location and the axis of symmetry.
• \( c = -46 \) gives a vertical shift to the entire graph, moving it lower on the y-axis.

These coefficients are crucial as they provide insights into the graph’s movement and shape, allowing you to predict and sketch the graph even before plotting individual points.