Problem 6 - Basic Technical Mathematics with Calculus

Question

Sketch the graph of each parabola by using only the vertex and the \(y\) -intercept. Check the graph using a calculator. \(s=2 t^{2}+8 t-5\)

Step-by-Step Solution

Answer

The parabola has a vertex at \((-2, -13)\) and a y-intercept at \((0, -5)\). It opens upwards.

Step 1: Identify the Coefficients

The given equation of the parabola is \( s = 2t^2 + 8t - 5 \). Identify the coefficients: \( a = 2 \), \( b = 8 \), and \( c = -5 \).

Step 2: Find the Vertex

Use the vertex formula \( t = -\frac{b}{2a} \) to find the \( t \)-coordinate of the vertex. Substitute \( a = 2 \) and \( b = 8 \) into the formula: \[ t = -\frac{8}{2 \times 2} = -2 \]. Then substitute \( t = -2 \) into the equation to find the \( s \)-coordinate: \[ s = 2(-2)^2 + 8(-2) - 5 = 8 - 16 - 5 = -13 \]. So, the vertex is \( (-2, -13) \).

Step 3: Find the Y-intercept

Set \( t = 0 \) in the equation to find the \( y \)-intercept: \[ s = 2(0)^2 + 8(0) - 5 = -5 \]. Thus, the y-intercept is \( (0, -5) \).

Step 4: Sketch the Parabola

Using the vertex \( (-2, -13) \) and the \( y \)-intercept \( (0, -5) \), sketch the parabola. Since \( a = 2 \), the parabola opens upwards. Plot these points on a coordinate plane and draw a smooth curve through them.

Step 5: Check with Calculator

Use a graphing calculator to input \( s = 2t^2 + 8t - 5 \). Compare the calculator's graph with your sketch to ensure accuracy. Both should have the same vertex and \( y \)-intercept, confirming the correct shape and orientation.

Key Concepts

Vertex Calculation

The vertex of a parabola is a crucial point that shows its maximum or minimum value. To find it in the equation given as \( s = 2t^2 + 8t - 5 \), we utilize a specific formula:

• The vertex formula is \( t = -\frac{b}{2a} \).
• In our scenario, the coefficients are \( a = 2 \) and \( b = 8 \).

Substituting in these values, the calculation becomes \( t = -\frac{8}{2 \times 2} = -2 \). Now, we have the \( t \)-coordinate of the vertex.
To find the corresponding \( s \)-value, substitute \( t = -2 \) back into the original quadratic equation:

• \( s = 2(-2)^2 + 8(-2) - 5 \)
• This simplifies the computation to \( 8 - 16 - 5 = -13 \).

Hence, the vertex for this parabola is \( (-2, -13) \). This point reveals that the parabola takes its minimum value since \( a > 0 \) and opens upwards.

Y-Intercept Identification

Finding the y-intercept is straightforward. It simply involves setting \( t = 0 \) in the quadratic equation \( s = 2t^2 + 8t - 5 \). This step calculates where the parabola crosses the \( y \)-axis. Here's how to do it:

• Setting \( t = 0 \) gives: \( s = 2(0)^2 + 8(0) - 5 \).
• The result of the computation is \( s = -5 \).

This indicates the y-intercept is \( (0, -5) \).
The y-intercept is a point where the parabola intersects the \( y \)-axis, providing insight into its position relative to the origin of the coordinate plane.

Quadratic Equation

Quadratic equations, like the one in our example \( s = 2t^2 + 8t - 5 \), are polynomial equations of degree two. They are fundamental to algebra and often appear in various real-world contexts. Consider the structure of a quadratic equation:

• General form: \( ax^2 + bx + c = 0 \)
• Our equation is presented as \( s = 2t^2 + 8t - 5 \).

Each quadratic equation has a distinct

• Apex: Known as the vertex, the point where the curve is at its maximum or minimum.
• Symmetry: The parabola is symmetric about a vertical line running through the vertex.
• Solution: Can be found using various methods such as factoring, completing the square, or the quadratic formula.

Understanding the elements of quadratic equations aids in predicting the function's behavior, sketching the curve correctly, and finding additional points like intercepts.

Graphing Calculator

A graphing calculator is an indispensable tool for visualizing equations and verifying manual sketches. Here's how you can use this tool for our quadratic equation \( s = 2t^2 + 8t - 5 \):

• Input the full equation into the calculator.
• Access the graphing feature to generate the curve.
• Observe key coordinates, ensuring the vertex is \( (-2, -13) \) and the y-intercept \( (0, -5) \).

Using a graphing calculator helps confirm the key properties of the parabola, like the direction it opens (upward for \( a = 2 \)), the accuracy of plotted points, and the overall shape of the graph.
It's a powerful way to ensure accuracy in sketches and understand the function thoroughly, acting as a checkpoint alongside manual computations.