/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Sketch the graph of each parabol... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 15

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. $$y=-2 x^{2}-2 x-6$$

Short Answer

Expert verified
Vertex: \( \left(\frac{1}{2}, -\frac{9}{2}\right) \), \( y \)-intercept: \( (0, -6) \), Other points: \( (-1, -6) \), \( (1, -10) \); use to sketch the parabola.

Step by step solution

01

Determine the Vertex

To find the vertex of the parabola, use the vertex formula for a parabola in the form of \( y = ax^2 + bx + c \). The vertex \( x \)-coordinate is given by \( x = -\frac{b}{2a} \). Here, \( a = -2 \), \( b = -2 \), so \( x = -\frac{-2}{2(-2)} = \frac{1}{2} \). Substitute back into the equation to find the \( y \)-coordinate: \( y = -2\left(\frac{1}{2}\right)^2 - 2\left(\frac{1}{2}\right) - 6 = -\frac{1}{2} - 1 - 6 = -\frac{9}{2} \). Therefore, the vertex is \( \left(\frac{1}{2}, -\frac{9}{2}\right) \).
02

Find the y-intercept

The \( y \)-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the equation: \( y = -2(0)^2 - 2(0) - 6 = -6 \). Therefore, the \( y \)-intercept is \( (0, -6) \).
03

Locate Two Additional Points

Choose two \( x \)-values to find corresponding \( y \)-values. Let's choose \( x = -1 \) and \( x = 1 \). For \( x = -1 \): \( y = -2(-1)^2 - 2(-1) - 6 = -2 + 2 - 6 = -6 \). Thus, point \( (-1, -6) \). For \( x = 1 \): \( y = -2(1)^2 - 2(1) - 6 = -2 - 2 - 6 = -10 \). Thus, point \( (1, -10) \).
04

Graph the Parabola

Using the vertex \( \left(\frac{1}{2}, -\frac{9}{2}\right) \), the \( y \)-intercept \( (0, -6) \), and additional points \( (-1, -6) \) and \( (1, -10) \), plot these points on a graph. Sketch the parabola, ensuring it opens downwards due to the negative sign in \( ax^2 \).
05

Check the Graph on Calculator

Input the equation \( y = -2x^2 - 2x - 6 \) into a graphing calculator to verify the shape and position of the parabola. Ensure the plotted points and the directrix match, serving as a confirmation of the manual graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex of a Parabola
The vertex of a parabola is a crucial point on its graph. It's the peak or the lowest point, known as the maximum or minimum depending on the parabola's orientation. When a parabola is expressed in the form \( y = ax^2 + bx + c \), you can determine the vertex using the formula for the x-coordinate:
  • \( x = -\frac{b}{2a} \)
In our case, \( a = -2 \) and \( b = -2 \). By substituting these values, we find:
  • \( x = -\frac{-2}{2(-2)} = \frac{1}{2} \)
This gives the x-coordinate of the vertex. To find the y-coordinate, substitute \( x = \frac{1}{2} \) back into the equation:
  • \( y = -2\left(\frac{1}{2}\right)^2 - 2\left(\frac{1}{2}\right) - 6 = -\frac{9}{2} \)
Thus, the vertex of this parabola is \( \left(\frac{1}{2}, -\frac{9}{2}\right) \). This point is essential as it helps in drawing and understanding the parabola's orientation.
Y-Intercept
The y-intercept is where the graph of the parabola crosses the y-axis. It's the point where \( x = 0 \). To find the y-intercept of a parabola, substitute \( x = 0 \) into the equation and solve for \( y \):
  • \( y = -2(0)^2 - 2(0) - 6 = -6 \)
Hence, the y-intercept for our equation is \( (0, -6) \). This point is straightforward to find and is particularly useful when plotting because it gives a quick check point on the y-axis.
By locating the y-intercept, students can have one of the foundational points needed to sketch their parabola.
Additional Points on a Graph
Finding additional points is helpful for accurately sketching the parabola. Besides the vertex and y-intercept, selecting other x-values aids in depicting the curve of the parabola. Here鈥檚 how we find two more points:
  • Choose an x-value, say \( x = -1 \). Calculate the corresponding \( y \): \( y = -2(-1)^2 - 2(-1) - 6 = -6 \). So we get the point \((-1, -6)\).
  • Try another x-value, such as \( x = 1 \). Calculate: \( y = -2(1)^2 - 2(1) - 6 = -10 \). This gives the point \((1, -10)\).
Identifying these additional points allows you to see how the parabola behaves around its vertex. It also helps in confirming the parabolic shape, ensuring you plot a smooth curve through the calculated points.
Graphing Calculator Verification
Using a graphing calculator is a modern approach to graph verification. It allows students to ensure their manual plots are accurate by comparing them with calculator-generated graphs. Input the equation \( y = -2x^2 - 2x - 6 \) into the calculator; this will display a parabola, enabling you to:
  • Check that the vertex appears at \( \left(\frac{1}{2}, -\frac{9}{2}\right) \).
  • Verify the y-intercept occurs at \( (0, -6) \).
  • Ensure additional points like \( (-1, -6) \) and \( (1, -10) \) are correct.
This process is invaluable for students, as it visually reinforces the understanding of the parabola's geometry and assists in fixing any errors made during manual graphing.

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