Problem 84 - College Algebra

Question

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. $$y= 5 x^{2}+40 x+600$$

Step-by-Step Solution

Answer

The vertex of the function \(y = 5x^2 + 40x + 600\) is (-4, 400). A reasonable viewing rectangle for this function is \(x = [-14, 6]\) and \(y = [350, 450]\).

Step 1: Identify a, b, and c

In the given function, identify the coefficients a, b, c. For the function \(y = 5x^2 + 40x + 600\), \(a = 5\), \(b = 40\), and \(c = 600\).

Step 2: Find the x-coordinate of the vertex

Use the formula \(-b/2a\) to find the x-coordinate of the vertex. Substituting the given a and b values gives the result as \(-40/(2*5) = -4\)

Step 3: Find the y-coordinate of the vertex

Substitute \(x = -4\) into the original function to find the y-coordinate. Substituting, we get \(y = 5(-4)^2 + 40*(-4) + 600 = 400\).

Step 4: Determine a viewing rectangle

After finding the vertex (-4, 400), the minimum value of the function is the y-coordinate of the vertex i.e. 400. A reasonable viewing rectangle can extend 10 units to the left and right of x=-4 and from y = 350 (slightly below the minimum value) to y = 450 to capture a portion of the graph above and below the vertex.

Step 5: Graph the function

Graph the function using the vertex and the viewing rectangle determined above. Note that the function opens upwards because the coefficient of \(x^2\) is positive.

Key Concepts

Parabola

A parabola is a U-shaped graph that represents a quadratic function. The standard form of a quadratic equation is given by \( y = ax^2 + bx + c \). Depending on the coefficient \( a \), the parabola can open upwards or downwards. In our case with \( a = 5 \), the parabola opens upwards. This happens because a positive \( a \) value makes the ends of the graph rise.

The symmetry of a parabola means that it looks the same on both sides of a vertical line. This line of symmetry is crucial when finding the vertex.

Vertex

The vertex of a parabola is its peak or lowest point, depending on the direction it opens.

For a parabola that opens upwards, the vertex is the minimum point. Conversely, it is the maximum point for a downward-opening parabola.

In quadratic functions, the vertex can be found using the formula for the x-coordinate:

• \( x = \frac{-b}{2a} \)

This gives the x-value of the vertex.

For our function \( y = 5x^2 + 40x + 600 \), the x-coordinate is -4 from calculating \(-\frac{40}{2 \times 5}\).

By substituting this back into the function, we find the y-coordinate: 400, making the vertex \((-4, 400)\). This point is central to graphing the function and determining the parabola's shape.

Graphing Quadratic Functions

Graphing quadratic functions involves plotting their parabolas to visualize the function's behavior.

Start by plotting the vertex, which in our example is \((-4, 400)\). This is the point of symmetry and the lowest point for upward-opening parabolas.

Once the vertex is plotted, you can draw the parabola by considering the symmetric nature around the vertex. Quadratic graphs are symmetric, so you can use points equidistant from the vertex on both left and right sides to plot the curve accurately.

Additionally, noticing the direction of opening (upwards in this case) ensures correct sketching of the parabola's path.

Viewing Rectangle

A viewing rectangle helps in capturing important parts of the graph while using a graphing utility.

The viewing rectangle defines the visible area on your graphing calculator or software. It should cover regions of interest like vertices and intercepts.

For our problem, a sensible viewing rectangle extends 10 units left and right from the x-coordinate of the vertex (\(-4, 400\)). So, it ranges from \(x = -14\) to \(x = 6\). In the y-direction, we can opt for a range from 350 to 450, covering slightly below and above the vertex point.

This strategic selection of the window ensures that the graph is displayed clearly, capturing essential details while boosting understanding of the quadratic function's behavior.