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Chapter 0: Problem 60

Use a graphing utility to graph the equation. Identify any intercepts and test for symmetry. $$3 x-4 y^{2}=8$$

Short Answer

Expert verified
The graph of the given equation has an x-intercept at \(x = 8/3\) and no real-valued y-intercept. The graph is not symmetric with respect to the y-axis, x-axis or origin.

Step by step solution

01

Finding x-intercept

Set y=0 in the equation \(3 x-4 y^{2}=8\) to find the x-intercept. Hence the equation becomes \(3x = 8\), which simplifies to \(x = 8/3\). Therefore, the x-intercept of the graph is \(8/3\).
02

Finding y-intercept

Set x=0 in the equation \(3 x-4 y^{2}=8\) to find the y-intercept. Hence the equation becomes \(-4y^2 = 8\), which simplifies to \(y^2 = -2\). Therefore, there is no real-valued y-intercept for this equation as the square root of negative number is not defined in the real number system.
03

Test for symmetry

Replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation. The equation becomes \(-3x - 4y^2 = 8\), which is not identical to original equation. Therefore the graph of this equation is not symmetric with respect to y-axis, x-axis or the origin.

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Key Concepts

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

X-Intercept Calculation
Understanding how to calculate the x-intercept of a quadratic equation is essential for graphing and analyzing the behavior of the function. The x-intercept is the point where the graph of the equation crosses the x-axis, which means that the y-coordinate at this point is zero. To calculate the x-intercept of the equation \(3 x - 4 y^2 = 8\), set \(y=0\) and solve for \(x\).

This simplifies the equation to \(3x = 8\), which can be further reduced to \(x = \frac{8}{3}\). This means the graph of the equation crosses the x-axis at the point \((\frac{8}{3}, 0)\). This method can be applied to any quadratic function to find its x-intercepts, and it's crucial for sketching the graph accurately.
Y-Intercept Calculation
The y-intercept of a quadratic equation is where the graph intersects the y-axis. This occurs when \(x=0\). Determining the y-intercept provides insight into the starting point of the graph on the y-axis. For the equation \(3 x - 4 y^2 = 8\), setting \(x=0\) simplifies to \(-4y^2 = 8\).

However, a complication arises as the equation then becomes \(y^2 = -2\), which does not have a real solution since the square root of a negative number is not a real number. This indicates that the graph does not intersect the y-axis at any point. It's important for students to recognize situations where a quadratic graph does not have a real-valued y-intercept, as it will affect their understanding and representation of the graph.
Symmetry Testing
Symmetry in graphs allow us to predict behavior and attributes of the quadratic function. Testing for symmetry is a procedure where you substitute \(-x\) for \(x\) and \(-y\) for \(y\) to see if the equation remains unchanged. If it does, the graph is symmetric with respect to the origin, y-axis, or x-axis, depending on the type of substitution results.

For the equation \(3 x - 4 y^2 = 8\), replacing \(x\) with \(-x\) leads to \(-3x - 4y^2 = 8\), which is not the same as the original equation. This shows us that the graph does not have symmetry along the y-axis, nor does it have origin symmetry. Similarly, replacing \(y\) with \(-y\) yields the same equation because \(y^2\) is unchanged by the sign of \(y\), indicating symmetry with respect to the x-axis. Testing for symmetry is a valuable tool for visualizing graphs and understanding their behavior without plotting numerous points.

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Most popular questions from this chapter

Apartment Rental A real estate office handles an apartment complex with 50 units. When the rent is $$\$ 580$$ per month, all 50 units are occupied. However, when the rent is $$\$ 625$$, the average number of occupied units drops to 47 . Assume that the relationship between the monthly rent \(p\) and the demand \(x\) is linear. (Note: The term demand refers to the number of occupied units.) (a) Write a linear equation giving the demand \(x\) in terms of the rent \(p\). (b) Linear extrapolation Use a graphing utility to graph the demand equation and use the trace feature to predict the number of units occupied if the rent is raised to $$\$ 655$$. (c) Linear interpolation Predict the number of units occupied if the rent is lowered to $$\$ 595$$. Verify graphically.

Automobile Aerodynamies The horsepower \(H\) required to overcome wind drag on a certain automobile is approximated by \(H(x)=0.002 x^{2}+0.005 x-0.029, \quad 10 \leq x \leq 100\) where \(x\) is the speed of the car in miles per hour. (a) Use a graphing utility to graph \(H\). (b) Rewrite the power function so that \(x\) represents the speed in kilometers per hour. [Find \(H(x / 1.6) .]\)

Find the domain of the function. $$g(x)=\frac{2}{1-\cos x}$$

Volume An open box of maximum volume is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure). (a) Write the volume \(V\) as a function of \(x\), the length of the corner squares. What is the domain of the function? (b) Use a graphing utility to graph the volume function and approximate the dimensions of the box that yield a maximum volume. (c) Use the table feature of a graphing utility to verify your answer in part (b). (The first two rows of the table are shown.) $$ \begin{array}{|c|c|c|} \hline \text { Height, } x & \begin{array}{c} \text { Length } \\ \text { and Width } \end{array} & \text { Volume, } \boldsymbol{V} \\ \hline 1 & 24-2(1) & 1[24-2(1)]^{2}=484 \\ \hline 2 & 24-2(2) & 2[24-2(2)]^{2}=800 \\ \hline \end{array} $$

Find the domain and range of the function. $$h(x)=-\sqrt{x+3}$$
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