Problem 62 - 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 parabola \(y = 5x^{2} + 40x + 600\) is (-4, 440). The recommended viewing rectangle for graphing this quadratic function is from x = -10 to x = 5 and y = 0 to y = 500.

Step 1: Identify the coefficients

From the quadratic function \(y = 5x^{2} + 40x + 600\), identify the coefficients a, b, and c. In this case, a is 5, b is 40 and c is 600.

Step 2: Compute the x-coordinate of the vertex

Use the formula \(-b/(2a)\) to find the x-coordinate of the vertex. In this example, it will be \(-40/(2*5) = -4\).

Step 3: Compute the y-coordinate of the vertex

Substitute the x-coordinate of the vertex into the function to find the y-coordinate. Therefore, \(y = 5*(-4)^{2} + 40*(-4) + 600 = 440\). Thus, the vertex is (-4, 440).

Step 4: Determine Reasonable Viewing Rectangle

To provide a clear view, focus on the vertex, a possible viewing window would be a rectangle where the x- and y- intervals are slight extensions from -4 and 440 respectively. This may be from x = -10 to x = 5 (a bit wider than the vertex x-coordinate -4 for an adequate viewpoint), and y = 0 to y = 500 (going up to a bit more than the vertex y-coordinate 440).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a key feature that represents either the highest or lowest point of the curve, depending on its orientation. This is determined by the quadratic function. In the function provided, \(y = 5x^2 + 40x + 600\), the vertex can be found using a simple mathematical formula.
The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Here, "b" and "a" are coefficients from the equation. For our function, \(a = 5\) and \(b = 40\). Substitute these values into the formula: \(x = -\frac{40}{2 \times 5} = -4\).

Next, to find the y-coordinate, substitute \( x = -4 \) back into the function \(y = 5(-4)^2 + 40(-4) + 600\). Upon simplification, the y-coordinate is 440. Thus, the vertex of the parabola is at \((-4, 440)\).
The vertex form of a quadratic equation is very useful when sketching the curve, as it provides a clear point of reference for the shape of the parabola.

Graphing Utility

Graphing utilities are powerful tools that help visualize functions, including quadratic functions like the one provided. With a graphing utility, students can graphically represent equations, explore different views, and verify calculated points like the vertex.
To effectively use a graphing utility, set up a reasonable viewing window. For our function where the vertex is \((-4, 440)\), a potential window could span more than what's necessary to clearly see the parabola's features.

• For the x-axis, you could choose a range from \(-10\) to \(5\). This extends slightly beyond the vertex x-coordinate \(-4\) to accommodate the full width of the parabola.
• For the y-axis, consider a range from \(0\) to \(500\), slightly above the vertex y-coordinate \(440\) to showcase the parabola's depth.

These constraints ensure that important features of the graph are visible, enhancing understanding through a clear visual representation.

Quadratic Equation Coefficients

Quadratic equation coefficients are crucial in determining the shape and position of a parabola. In the equation \(y = ax^2 + bx + c\), the values of \(a\), \(b\), and \(c\) influence the vertex, direction, and width of the parabola.
The coefficient \(a\) determines the direction in which the parabola opens. If \(a\) is positive, the parabola opens upwards, and if negative, it opens downwards. In our case, \(a = 5\), so the parabola opens upwards, indicating a minimum point at the vertex.
The coefficient \(b\) affects the position of the vertex along the x-axis. It, combined with \(a\), helps calculate the x-coordinate of the vertex using \(-\frac{b}{2a}\). Here, \(b = 40\), playing a role in locating \(x = -4\).
The constant term \(c\) represents the y-intercept, giving the point where the parabola crosses the y-axis. For our function, \(c = 600\), so the parabola intersects the y-axis at \((0, 600)\). Understanding these coefficients is key to graphing and analyzing the behavior of quadratic functions.