Problem 12 - Basic Technical Mathematics with Calculus

Question

Sketch the graph of each parabola by using the vertex, the \(y\) -intercept, and the \(x\) -intercepts. Check the graph using \(a\) calculator. \(y=-2 x^{2}-6 x+8\)

Step-by-Step Solution

Answer

Vertex: \(\left(\frac{3}{2}, -5.5\right)\), \(y\)-intercept: \((0, 8)\), \(x\)-intercepts: \((-4, 0)\), \((1, 0)\).

Step 1: Identify the Vertex

The vertex of a parabola in the form \(y = ax^2 + bx + c\) is found using the formula \(x = -\frac{b}{2a}\). For the equation \(y = -2x^2 - 6x + 8\), \(a = -2\) and \(b = -6\). Compute \(x = -\frac{-6}{2(-2)} = \frac{6}{4} = \frac{3}{2}\). Substitute \(x = \frac{3}{2}\) back into the equation to find \(y\):\[ y = -2\left(\frac{3}{2}\right)^2 - 6\left(\frac{3}{2}\right) + 8 = -2\cdot\frac{9}{4} - 9 + 8 = -\frac{18}{4} - 9 + 8 = -4.5 - 9 + 8 = -5.5.\] Thus, the vertex is \(\left(\frac{3}{2}, -5.5\right)\).

Step 2: Find the Y-Intercept

The \(y\)-intercept of a parabola occurs where \(x = 0\). Substitute \(x = 0\) into the equation \(y = -2x^2 - 6x + 8\):\[y = -2(0)^2 - 6(0) + 8 = 8.\] So, the \(y\)-intercept is \(0, 8\).

Step 3: Calculate the X-Intercepts

The \(x\)-intercepts are the points where the parabola crosses the \(x\)-axis, i.e., \(y = 0\). Set the equation to zero and solve for \(x\): \(-2x^2 - 6x + 8 = 0\). Simplified, this becomes \(x^2 + 3x - 4 = 0\). Factoring gives \((x + 4)(x - 1) = 0\), so the solutions are \(x = -4\) and \(x = 1\). Thus, the \(x\)-intercepts are \((-4, 0)\) and \((1, 0)\).

Step 4: Sketch the Graph

Using the vertex \(\left(\frac{3}{2}, -5.5\right)\), \(y\)-intercept \((0, 8)\), and \(x\)-intercepts \((-4, 0)\) and \((1, 0)\), you can sketch the graph. Start at the vertex and mark it on the coordinate plane. Plot the \(y\)-intercept and draw the parabola heading up through the \(x\)-intercepts, making sure it reflects the symmetric nature around the vertical line through the vertex.

Step 5: Verify with a Calculator

Use a graphing calculator to input the equation \(y = -2x^2 - 6x + 8\). Confirm that the graph on the calculator matches the sketch, ensuring accuracy of the intercepts and the vertex.

Key Concepts

Understanding the Vertex Formula

In graphing parabolas, the vertex is one of the most crucial points to identify. A parabola in the form \( y = ax^2 + bx + c \) has a vertex that can be found using the vertex formula:\[ x = -\frac{b}{2a} \]This formula helps us find the x-coordinate of the vertex. Once you have the x-coordinate, substitute it back into the original equation to find the corresponding y-coordinate.

• For example, consider the equation \( y = -2x^2 - 6x + 8 \).
• Identify \( a = -2 \) and \( b = -6 \).
• Apply the formula: \( x = -\frac{-6}{2(-2)} = \frac{3}{2} \).
• This x-value is then used to calculate the y-coordinate by substituting it back: \( y = -5.5 \).

Thus, the vertex for this particular parabola is \( \left( \frac{3}{2}, -5.5 \right) \). The vertex provides a point of symmetry and reflects the parabola's highest or lowest point, depending on the direction it opens.

Calculating X-Intercepts

The x-intercepts of a parabola are the points where the graph crosses the x-axis. Mathematically, these points occur where \( y = 0 \). To find the x-intercepts:

• Set the equation \( y = ax^2 + bx + c \) to zero. For our example: \( -2x^2 - 6x + 8 = 0 \).
• Rearrange this into a standard quadratic form: \( x^2 + 3x - 4 = 0 \).
• Solve using factoring: \( (x + 4)(x - 1) = 0 \), giving solutions \( x = -4 \) and \( x = 1 \).

Thus, the x-intercepts for this parabola are \( (-4, 0) \) and \( (1, 0) \). These points are crucial as they indicate where the curve changes direction with respect to the x-axis.

Identifying the Y-Intercept

The y-intercept provides a key reference point for graphing a parabola. It is where the graph crosses the y-axis and occurs at \( x = 0 \). To find the y-intercept:

• Substitute \( x = 0 \) into the equation \( y = -2x^2 - 6x + 8 \).
• This simplifies to \( y = 8 \).

Thus, the y-intercept for the parabola is \( (0, 8) \). This point is especially useful as it provides a starting point for sketching the graph along with understanding the general direction of the parabola.

Exploring Parabola Symmetry

Parabolas exhibit a unique symmetry around a vertical line known as the axis of symmetry. This line runs through the vertex and divides the parabola into two mirror-image halves.

• For the instance \( y = -2x^2 - 6x + 8 \), the axis of symmetry is given by \( x = \frac{3}{2} \).
• This x-coordinate represents the vertical line of symmetry.
• Graphically, if a point \( (x, y) \) lies on the parabola, there will be another point \( (2 \times \frac{3}{2} - x, y) \) equidistant from the line of symmetry.

Understanding symmetry helps in ensuring the parabola is accurately plotted, providing insight into its properties and ensuring consistency in plotting graph points.