Problem 36 - 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. $$P(x)=-\sqrt{2} x^{2}+0.45 x+1.39$$

Step-by-Step Solution

Answer

Vertex: (0.16, 1.43); X-intercepts: -0.54 and 1.24.

Step 1: Enter the Function

Enter the function \( P(x) = -\sqrt{2} x^2 + 0.45x + 1.39 \) into your graphing calculator. Choose a viewing window; a good starting point is \(-10 \) to \(10 \) for \(x\) and \(-10 \) to \(10 \) for \(y\).

Step 2: Find the Vertex

Use the calculator's graphing tool to find the vertex of the parabola. Access the 'Calculate' or 'Trace' function and locate the vertex, which is where the function reaches its maximum/minimum. For a downward opening parabola like this one, it's the maximum point.

Step 3: Record the Vertex Coordinates

Once the vertex is found using your calculator, record its coordinates. You might find coordinates approximately at \((0.16, 1.43)\) when the calculation is done correctly.

Step 4: Find the X-Intercepts

To find the \(x\)-intercepts, use the calculator function to find where the graph crosses the \(x\)-axis. This will require using the 'Zero' or 'Root' function on your calculator.

Step 5: Record the X-Intercepts

Identify the points where the graph intersects the \(x\)-axis. The approximate \(x\)-intercepts should be around \(-0.54\) and \(1.24\).

Step 6: Verify by Substitution

Verify the intercepts by substituting them back into the equation: \((-\sqrt{2})(x^2) + 0.45x + 1.39 = 0\). Confirm both solutions close to \(x \approx -0.54\) and \(x \approx 1.24\) solve the equation.

Key Concepts

Graphing Calculators

A graphing calculator is an essential tool for visualizing mathematical functions. It helps students understand the behavior and characteristics of functions, such as quadratic functions. For the given function, \( P(x) = -\sqrt{2} x^2 + 0.45x + 1.39 \), you first need to input this formula into the graphing calculator.
Choose an appropriate viewing window to see the complete shape of the parabola. A typical setup might range both axes from -10 to 10.
This ensures the central features, like the vertex and the x-intercepts, are visible.

• Enter the equation accurately.
• Adjust the window settings as needed for clarity.
• Use graphing tools to explore different parts of the graph.

Once inputted, the calculator provides a visual representation. This image makes complex equations more comprehensible and aids in manually approximating the vertex and x-intercepts.

Finding Vertex

The vertex of a quadratic function is a key point, showing the peak or valley of the parabola. In the case of a quadratic like \( P(x) = -\sqrt{2} x^2 + 0.45x + 1.39 \), it opens downwards, making the vertex the maximum.
To find it using a graphing calculator:

• Access the 'Calculate' or 'Trace' feature.
• Navigate to the highest point of the graph.
• The calculator may provide options to find maximum or minimum values.

The coordinates you find are essential. They are approximate, so expect them to vary slightly, for example, around \((0.16, 1.43)\). Record these values because they provide insights into the function's behavior.

X-Intercepts

X-intercepts are points where the graph crosses the x-axis. They represent the solutions to the quadratic equation when set equal to zero. For \( P(x) = -\sqrt{2} x^2 + 0.45x + 1.39 \), they are crucial for understanding where this parabola intersects the x-axis.
Finding these intercepts on a graphing calculator involves:

• Using the 'Zero' or 'Root' function in your calculator.
• Moving along the graph to identify the points where it crosses the x-axis.
• The calculator will give close approximate points.

For this function, the x-intercepts might be around \(-0.54\) and \(1.24\). These intercepts need verification by substitution back into the original equation, ensuring they satisfy \((-\sqrt{2})x^2 + 0.45x + 1.39 = 0\). Recognizing these solutions visually with a graphing calculator enhances understanding.