Problem 15 - Basic Technical Mathematics with Calculus

Question

Sketch the graph of each parabola by using the vertex, the \(y\) -intercept, and two other points, not including the \(x\) -intercepts. Check the graph using a calculator. \(y=-2 x^{2}-2 x-6\)

Step-by-Step Solution

Answer

The vertex is \((-\frac{1}{2}, -\frac{11}{2})\), the \(y\)-intercept is \((0, -6)\), and two additional points are \((-1, -6)\) and \((1, -10)\). The parabola opens downwards and is verified by graphing.

Step 1: Identify the Vertex

The equation of the parabola is given in the standard form: \[ y = ax^2 + bx + c \]where \( a = -2 \), \( b = -2 \), and \( c = -6 \). The vertex \((h, k)\) of a parabola in standard form can be found using:\[ h = \frac{-b}{2a} \]Substituting the values, we get:\[ h = \frac{-(-2)}{2(-2)} = \frac{2}{-4} = -\frac{1}{2} \]Next, substitute \( x = h = -\frac{1}{2} \) into the original equation to find \( k \):\[ y = -2\left(-\frac{1}{2}\right)^2 - 2\left(-\frac{1}{2}\right) - 6 = -2\left(\frac{1}{4}\right) + 1 - 6 = -\frac{1}{2} + 1 - 6 = -\frac{11}{2} \]Thus, the vertex is \((-\frac{1}{2}, -\frac{11}{2})\).

Step 2: Find the Y-intercept

The \( y \)-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the equation:\[ y = -2(0)^2 - 2(0) - 6 = -6 \]Thus, the \( y \)-intercept is \((0, -6)\).

Step 3: Find Two Additional Points

Choose two values of \( x \) to find additional points. For example, choose \( x = -1 \) and \( x = 1 \).For \( x = -1 \):\[ y = -2(-1)^2 - 2(-1) - 6 = -2(1) + 2 - 6 = -6 \]Thus, one point is \((-1, -6)\).For \( x = 1 \):\[ y = -2(1)^2 - 2(1) - 6 = -2 - 2 - 6 = -10 \]Thus, another point is \((1, -10)\).

Step 4: Sketch the Graph

Plot the vertex \((-\frac{1}{2}, -\frac{11}{2})\), the \(y\)-intercept \((0, -6)\), and the additional points \((-1, -6)\) and \((1, -10)\) on a coordinate plane. Connect these points with a smooth, parabolic curve that opens downwards since \( a = -2 < 0 \).

Step 5: Verify with a Calculator

Use a graphing calculator to check the accuracy of your sketch. Input the equation \( y = -2x^2 - 2x - 6 \) and compare the on-screen graph with your hand-drawn version to ensure they match.

Key Concepts

Vertex Identification

Identifying the vertex of a parabola is an important first step in graphing. For a quadratic equation in standard form, such as \( y = ax^2 + bx + c \), the vertex provides a crucial point of reference. The vertex

• is a turning point of the parabola.
• determines whether the parabola opens upwards or downwards depending on the sign of \( a \).

To find the vertex \((h, k)\), use the formula for \( h \):\[ h = \frac{-b}{2a} \]Plugging in the values from the equation \( y = -2x^2 - 2x - 6 \), we calculate:\[ h = \frac{-(-2)}{2(-2)} = -\frac{1}{2} \]
Then, substitute back to determine \( k \):\[ k = y(-\frac{1}{2}) = -2\left(-\frac{1}{2}\right)^2 - 2\left(-\frac{1}{2}\right) - 6 = -\frac{11}{2} \] Thus, the vertex is \((-\frac{1}{2}, -\frac{11}{2})\).This tells you the graph turns at this point and provides symmetry for plotting.

Y-Intercept

The y-intercept is where the graph of the parabola crosses the y-axis. This is a significant point because it is straightforward to determine and plot. To find the y-intercept, simply replace \( x \) with 0 in the equation. Thus for \( y = -2x^2 - 2x - 6 \), the calculation is:
\[ y = -2(0)^2 - 2(0) - 6 = -6 \]The y-intercept is therefore \((0, -6)\). This point tells us the value of \( y \) when \( x = 0 \), and it's directly plotted on the y-axis. The y-intercept gives one point to help draw the parabola and understand its intersection with the y-axis.

Coordinate Points

Additional coordinate points supplement the vertex and y-intercept when sketching a parabola. They give more structure to the graph you're about to draw. Choose any values for \( x \) to calculate corresponding \( y \) values.
1. When \( x = -1 \): \[ y = -2(-1)^2 - 2(-1) - 6 = -6 \] Point: \((-1, -6)\)
2. When \( x = 1 \): \[ y = -2(1)^2 - 2(1) - 6 = -10 \] Point: \((1, -10)\)These points provide additional frames for the graph and help make the shape more accurate. Placing them with the vertex and y-intercept helps create a fuller picture of the parabola, revealing its downward opening due to the negative \( a \) value in the equation.

Graphing Calculator Verification

Once you have plotted the vertex, y-intercept, and additional points, it's time to draw the parabola. But how do you ensure accuracy? Using a graphing calculator is a fantastic way to verify your work.
Here's how it works:

• Input the quadratic equation: \( y = -2x^2 - 2x - 6 \) into the calculator.
• The calculator generates the curve and displays it on the screen.
• Compare this with your hand-drawn version to see if they align.

Graphing calculators allow you to quickly check your work, ensuring fidelity to the mathematical model.They offer a powerful visual confirmation of your effort in sketching the parabola.