Problem 38 - 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. $$y=-0.55 x^{2}+3.21 x \quad$$

Step-by-Step Solution

Answer

Vertex: (2.92, 4.68); X-intercepts: (0, 0) and (5.84, 0).

Step 1: Identify the Function Type

The given function is a quadratic of the form \(y = ax^2 + bx + c\). In this case, \(a = -0.55\), \(b = 3.21\), and \(c = 0\). Quadratic functions have parabolic graphs that open upwards if \(a > 0\) and downwards if \(a < 0\). Here, since \(a < 0\), the parabola opens downwards.

Step 2: Calculate the Vertex

The vertex of a quadratic function \(y = ax^2 + bx + c\) can be found using the formula \(x = -\frac{b}{2a}\). Substituting the values, we get: \[x = -\frac{3.21}{2(-0.55)} = \frac{3.21}{1.1} = 2.9181\]. We now find the \(y\)-coordinate by substituting \(x = 2.9181\) back into the function: \(y = -0.55(2.9181)^2 + 3.21(2.9181)\). Calculating, we find \(y \approx 4.68\). Thus, the vertex is approximately \((2.92, 4.68)\).

Step 3: Find the X-Intercepts

The x-intercepts can be found by solving \(-0.55x^2 + 3.21x = 0\). Factor out an \(x\): \(x(-0.55x + 3.21) = 0\). Thus, the intercepts occur at \(x = 0\) and at \(-0.55x + 3.21 = 0\), giving \(x = \frac{3.21}{0.55}\), which is approximately \(x = 5.84\). So the x-intercepts are \((0, 0)\) and \((5.84, 0)\).

Step 4: Graph the Function

Use a graphing calculator to plot the parabola. Set the window to focus around the vertex and x-intercepts, ensuring you include the points from around \(x = 0\) to \(x = 6\) on the x-axis and from \(y = -1\) to \(y = 5\) on the y-axis. Verify the vertex and intercept calculations by observing the graph.

Key Concepts

Vertex Calculation

In quadratic functions, the vertex is a crucial point as it represents the peak or the lowest point of the graph. For the quadratic function given by \(y = ax^2 + bx + c\), the vertex \((h, k)\) can be calculated using the formula for the \(x\)-coordinate of the vertex: \(x = -\frac{b}{2a}\).

To find the \(y\)-coordinate \((k)\), simply substitute the \(x\)-value back into the quadratic equation. In our exercise, for \(a = -0.55\) and \(b = 3.21\), we plug these into our formula:

• 
  \[x = -\frac{3.21}{2(-0.55)} = \frac{3.21}{1.1} = 2.9181\]
  


• Then, substitute back to find \(y\), giving us: \[y = -0.55(2.9181)^2 + 3.21(2.9181)\]
  


• After computation, \(y \approx 4.68\), so the vertex is approximately \((2.92, 4.68)\).
  

Remember, when dealing with calculations, especially ones that require precision like finding the vertex, be careful with rounding at early stages to avoid inaccuracies later on.

X-Intercepts

The \(x\)-intercepts are where the graph of the quadratic function crosses the \(x\)-axis. These points occur when \(y = 0\). For the equation \(-0.55x^2 + 3.21x = 0\), you can solve for \(x\) by factoring.

Factoring out an \(x\) gives:

• \(x(-0.55x + 3.21) = 0\)


• Setting each factor equal to zero provides the solutions:


• \(x = 0\)


• \(-0.55x + 3.21 = 0\), solving gives \(x = \frac{3.21}{0.55} \approx 5.84\)
  

Thus, the \(x\)-intercepts of the parabola are at the points \((0, 0)\) and \((5.84, 0)\). To ensure accuracy, always double-check your factored equation and solutions.

This process involves careful algebraic manipulation, making it important to verify each step promptly.

Graphing Calculator

Using a graphing calculator can simplify the process of sketching quadratic functions and verifying calculations like the vertex and x-intercepts. Here, it's essential to set an appropriate viewing window to capture critical points.

For our function, we suggest setting your calculator's window to:

• \(x\)-axis: from approximately \(x = 0\) to \(x = 6\), since the intercepts and vertex fall within this range.
  


• \(y\)-axis: from approximately \(y = -1\) to \(y = 5\), allowing you to capture the turning point of the parabola.
  

Once your window is set, graph the function:

• Ensure that the parabola opens downwards, confirming the negative coefficient \(a = -0.55\).
  


• Verify the vertex and open up the trace function (if available) to check coordinates aproximations.
  


• Use plots to reflect x-intercepts \((0, 0)\) and \((5.84, 0)\).
  

Understanding how to effectively use a graphing calculator will enhance your problem-solving skills and give visual confirmation of your solutions.