Problem 35 - A Graphical Approach to College Algebra

Question

Graph each function in a viewing window that will allow you to use your calculator to approximate (a) the coordinates of the vertex and (b) the \(x\) -intercepts. Give values to the nearest hundredth. $$P(x)=-0.32 x^{2}+\sqrt{3} x+2.86$$

Step-by-Step Solution

Answer

Vertex: (2.70, 5.81), x-intercepts: (-2.23, 0) and (3.97, 0).

Step 1: Identify the function

The given function is a quadratic function: \[ P(x) = -0.32x^2 + \sqrt{3}x + 2.86 \]This function is in the standard form \( ax^2 + bx + c \). Identify \( a = -0.32 \), \( b = \sqrt{3} \), and \( c = 2.86 \).

Step 2: Determine the vertex

To find the vertex of a quadratic function in standard form, use the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a} \]Substitute \( a = -0.32 \) and \( b = \sqrt{3} \):\[ x = -\frac{\sqrt{3}}{2(-0.32)} \approx 2.70 \]Next, substitute \( x = 2.70 \) back into the function to find the y-coordinate of the vertex:\[ P(2.70) = -0.32(2.70)^2 + \sqrt{3}(2.70) + 2.86 \approx 5.81 \]The vertex is approximately \((2.70, 5.81)\).

Step 3: Determine the x-intercepts

To find the \( x\)-intercepts, set \( P(x) = 0 \) and solve for \( x \): \[ -0.32x^2 + \sqrt{3}x + 2.86 = 0 \]Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -0.32 \), \( b = \sqrt{3} \), \( c = 2.86 \).First, calculate the discriminant:\[ b^2 - 4ac = (\sqrt{3})^2 - 4(-0.32)(2.86) \approx 9.61 \]Since the discriminant is positive, there are two real solutions:\[ x = \frac{ -\sqrt{3} \pm \sqrt{9.61} }{2(-0.32)} \]Calculate the solutions:\[ x_1 \approx -2.23, \quad x_2 \approx 3.97 \]So, the x-intercepts are approximately \((-2.23, 0)\) and \((3.97, 0)\).

Step 4: Plot the function on a graphing calculator

Set up your calculator with an appropriate viewing window based on the vertex and x-intercepts:- Xmin: -5 - Xmax: 5- Ymin: -1 - Ymax: 10Enter the function \( P(x) = -0.32x^2 + \sqrt{3}x + 2.86 \) and graph it to verify the appearing vertex at about \((2.70, 5.81)\) and the x-intercepts near \((-2.23, 0)\) and \((3.97, 0)\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a key feature that tells us about the minimum or maximum point of the parabola. Since the function given is \[ P(x) = -0.32x^2 + \sqrt{3}x + 2.86 \]and has a negative coefficient for the \(x^2\) term (\(-0.32\)), this parabola will open downward, indicating a maximum point at the vertex.
To find the vertex of the parabola when the function is in standard form, use the vertex formula:\[ x = -\frac{b}{2a} \]
This formula provides the \(x\)-coordinate of the vertex. After substituting \(a = -0.32\) and \(b = \sqrt{3}\), we find:\[ x \approx 2.70 \]Substituting \(x = 2.70\) back into the original equation allows us to calculate the \(y\)-coordinate:\[ P(2.70) \approx 5.81 \]
Thus, the vertex of the parabola is approximately \((2.70, 5.81)\). This provides insights into the graph, like where it reaches its peak.

X-Intercepts

X-intercepts are points where the parabola crosses the \(x\)-axis. At these points, the function value \(P(x)\) is zero. To find the \(x\)-intercepts, we set:\[ -0.32x^2 + \sqrt{3}x + 2.86 = 0 \]and solve the equation for \(x\).
The quadratic formula helps us find these intercepts:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Using \(a = -0.32\), \(b = \sqrt{3}\), and \(c = 2.86\), we calculate the discriminant:\[ b^2 - 4ac \approx 9.61 \]
Note: A positive discriminant indicates two real solutions.

• The first solution: \(x \approx -2.23\)
• The second solution: \(x \approx 3.97\)

Therefore, the \(x\)-intercepts are approximately \((-2.23, 0)\) and \((3.97, 0)\). These values signify where the parabola touches and crosses the \(x\)-axis, highlighting important horizontal limits of the graph.

Quadratic Formula

The Quadratic Formula is a crucial tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). This formula helps us find the values of \(x\) when the equation equates to zero. The formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula includes several important components:

• \(b^2 - 4ac\) is the discriminant which tells us the nature of the roots (real and distinct, real and repeated, or complex).
• The \(\pm\) symbol indicates there may be two solutions or intercepts for the equation.

For the problem at hand, by using \(a = -0.32\), \(b = \sqrt{3}\), and \(c = 2.86\), we specifically calculated the discriminant to be \(9.61\). Since the discriminant is positive, the quadratic equation has two real solutions, which correspond to the parabola's \(x\)-intercepts found earlier. The Quadratic Formula simplifies the process of finding these intercepts, providing precise solutions in complex cases where factoring is challenging.