Problem 28 - 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. $$P(x)=-x^{2}+2 x+1$$

Step-by-Step Solution

Answer

The vertex is (1, 2) and the parabola opens downward.

Step 1: Identify the Quadratic Function Components

The given quadratic function is in the standard form, \( P(x) = ax^2 + bx + c \). Here, \( a = -1 \), \( b = 2 \), and \( c = 1 \).

Step 2: Apply the Vertex Formula

The vertex \( (h, k) \) of a quadratic function \( ax^2 + bx + c \) is found using the formulas \( h = -\frac{b}{2a} \) and \( k = P(h) \). Substitute \( a = -1 \) and \( b = 2 \) to find \( h \): \[ h = -\frac{2}{2 imes (-1)} = 1 \]

Step 3: Calculate the Vertex y-coordinate

Substitute \( x = 1 \) back into the original function \( P(x) = -x^2 + 2x + 1 \) to find \( k \): \[ k = P(1) = -1^2 + 2(1) + 1 = -1 + 2 + 1 = 2 \]Thus, the vertex is \( (1, 2) \).

Step 4: Sketch the Parabola

Since the coefficient \( a = -1 \) is negative, the parabola opens downward. Plot the vertex \((1, 2)\) on the graph. To confirm the shape, find additional points, such as by evaluating \( P(x) \) at \( x = 0 \) and \( x = 2 \). For \( x=0 \), \( P(0) = 1 \), and for \( x=2 \), \( P(2) = -1 \). Plot these points to guide your sketch of the downward-opening parabola.

Key Concepts

Vertex Formula

The vertex formula is a key tool in determining the vertex of a quadratic function. The vertex is a point that represents either the maximum or minimum value of the function depending on its orientation. The vertex of a quadratic function of the form \( ax^2 + bx + c \) is given by the coordinates \((h, k)\). To find \(h\), we use the formula:

• \( h = -\frac{b}{2a} \)

Once we have \(h\), we substitute it back into the function to find \(k\):

• \( k = P(h) \)

In the exercise, with \( P(x) = -x^2 + 2x + 1 \), we computed \( h = 1 \) and then found \( k = 2 \), giving us the vertex \((1, 2)\). Understanding how to find the vertex allows us to better visualize and graph the quadratic equation.

Parabola

A parabola is the graph of a quadratic function. It has a distinct U-shape, which can open upwards or downwards depending on the sign of the coefficient \(a\). In general, key characteristics of a parabola include:

• The vertex, which is its highest or lowest point.
• The axis of symmetry, a vertical line that passes through the vertex and divides the parabola into two mirror-image halves.
• The direction of the opening, which is upward if \(a > 0\) and downward if \(a < 0\).

For the function \( P(x) = -x^2 + 2x + 1 \), the graph is a parabola that opens downward since \(a = -1\). This means the vertex \((1, 2)\) is the maximum point on the graph.

Graphing Quadratics

Graphing quadratic functions involves several steps that help in sketching the shape accurately. Here's a straightforward process:

• First, determine the vertex using the vertex formula.
• Plot the vertex on a coordinate plane.
• Identify the direction in which the parabola opens by looking at the sign of \(a\).
• Calculate additional points around the vertex to define the shape.
• Use symmetry about the vertex to plot mirrored points on both sides of the parabola.

In the example \( P(x) = -x^2 + 2x + 1 \), after plotting the vertex \((1, 2)\), we calculated and plotted additional points such as \((0, 1)\) and \((2, -1)\) to guide the neat sketch of the parabola opening downward.

Standard Form of Quadratics

The standard form of a quadratic function is expressed as \( ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants. This form is very useful for identifying the quadratic's coefficient for determining properties like the direction of opening and the vertex's location:

• \(a\) gives us the parabola's orientation: if \(a > 0\), it opens upwards; if \(a < 0\), it opens downwards.
• \(b\) and \(c\) influence the specific position of the vertex and other points.

The given exercise function \( P(x) = -x^2 + 2x + 1 \) is directly in standard form, allowing us to immediately identify \(a = -1\), \(b = 2\), and \(c = 1\). This helps us apply methods like the vertex formula easily, guiding us through graphing and analyzing the function.