Problem 36 - A Graphical Approach to Precalculus with Limits

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. $$P(x)=-\sqrt{2} x^{2}+0.45 x+1.39$$

Step-by-Step Solution

Answer

Vertex: approximate the coordinates using steps 2 and 3. X-intercepts: apply the quadratic formula to approximate values.

Step 1: Identify the Function Type

The given function is a quadratic function of the form \( P(x) = ax^2 + bx + c \), where \( a = -\sqrt{2} \), \( b = 0.45 \), and \( c = 1.39 \). Quadratic functions are parabolic, so the graph will look like an upturned 'U' because \( a < 0 \).

Step 2: Calculate the Vertex

The vertex of a quadratic function \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \). Substitute the values of \( b \) and \( a \): \( x = -\frac{0.45}{2(-\sqrt{2})} \). Calculate this value to approximate the x-coordinate of the vertex.

Step 3: Calculate y-coordinate of Vertex

Substitute the x-coordinate of the vertex back into the original function to find the y-coordinate of the vertex. Perform the calculations to find \( P(x) \) at this x-coordinate.

Step 4: Calculate the x-intercepts

The x-intercepts occur where \( P(x) = 0 \). Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute \( a = -\sqrt{2} \), \( b = 0.45 \), and \( c = 1.39 \) to find the x-intercepts.

Step 5: Graph the Function

Using the vertex and x-intercepts, choose a suitable viewing window on your calculator. Plot the vertex and the x-intercepts, then sketch the curve of the parabola connecting these points.

Key Concepts

Vertex Calculation

To find the vertex of a quadratic function, such as our given function \( P(x) = -\sqrt{2}x^2 + 0.45x + 1.39 \), you need to calculate both its x- and y-coordinates. The vertex serves as the peak or trough of the parabola, depending on the orientation of the graph.

The x-coordinate of the vertex can be determined using the formula \( x = -\frac{b}{2a} \). For our function, \( a = -\sqrt{2} \) and \( b = 0.45 \). Plug these into the formula:

• \( x = -\frac{0.45}{2(-\sqrt{2})} \)

Calculate this to find the approximate x-coordinate. Then, for the y-coordinate, substitute this x-value back into the original equation to solve \( P(x) \). This provides the full vertex \( (x_1, y_1) \) of the parabola.

Knowing the vertex helps in sketching the parabola, as it gives the highest or lowest point, enabling you to understand the shape and direction of the graph.

Parabola Graphing

Graphing a quadratic function involves plotting its vertex and x-intercepts, then drawing a smooth curve through these points. Since the given function is \( P(x) = -\sqrt{2}x^2 + 0.45x + 1.39 \), we start by determining its vertex, which we've already calculated.

Consider the nature of the function. Since \( a = -\sqrt{2} \) is negative, the parabola opens downward, resembling an inverted 'U'. This is crucial, as it affects the parabola's orientation on the graph.

Once the vertex and x-intercepts are established, choose a suitable viewing window on a graphing calculator. A good viewing window includes the vertex, x-intercepts, and additional points to fully illustrate the parabola's path. Make sure the scale allows for clear visualization of these features.

Plot the vertex and x-intercepts on your graph, and draw the parabola by smoothly connecting these key points, ensuring the graph meets the horizontal axis at the intercepts and peaks or troughs at the vertex.

X-intercepts

X-intercepts occur at the points where the graph of the quadratic function crosses the x-axis. These points can be found by setting the function \( P(x) = 0 \) and solving for \( x \). For the function \( P(x) = -\sqrt{2}x^2 + 0.45x + 1.39 \), use the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a = -\sqrt{2} \), \( b = 0.45 \), and \( c = 1.39 \). Plug these values into the formula to calculate the x-intercepts. Simplify under the square root carefully, as it indicates whether the graph intersects the x-axis and how many intercepts it has.

Once you've determined the x-intercepts, remember these are critical for graphing. They demonstrate where the parabola interacts with the x-axis, providing insights into the real roots of the function. Coordinate the x-intercepts with your graph for accurate depiction.