Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 70
Make a table of values, and sketch the graph of the equation. Find the x- and y-intercepts, and test for symmetry. (a) \(y=-\sqrt{4-x^{2}}\) (b) \(x=y^{3}\)
Short Answer
Expert verified
(a) y-intercept: (0, -2); x-intercepts: (-2,0),(2,0); y-axis symmetry.
(b) Intercept: (0,0); origin symmetry.
Step by step solution
01
Determine values for x and calculate corresponding y-values (a)
For the equation \( y = -\sqrt{4-x^2} \), identify values of \( x \) which make the expression \( 4-x^2 \) non-negative. This is where \( -2 \leq x \leq 2 \). Compute values of \( y \) based on chosen \( x \)-values, such as \( x = -2, -1, 0, 1, 2 \). This gives corresponding \( y \)-values:- \( x = -2 \), \( y = -\sqrt{4 - (-2)^2} = 0 \)- \( x = -1 \), \( y = -\sqrt{4 - (-1)^2} = -\sqrt{3} \)- \( x = 0 \), \( y = -\sqrt{4} = -2 \)- \( x = 1 \), \( y = -\sqrt{4 - 1^2} = -\sqrt{3} \)- \( x = 2 \), \( y = -\sqrt{4 - 2^2} = 0 \).
02
Determine values for y and calculate corresponding x-values (b)
For the equation \( x = y^3 \), choose a range of \( y \) values and compute the corresponding \( x \)-values. For instance:- \( y = -2 \), \( x = (-2)^3 = -8 \)- \( y = -1 \), \( x = (-1)^3 = -1 \)- \( y = 0 \), \( x = (0)^3 = 0 \)- \( y = 1 \), \( x = (1)^3 = 1 \)- \( y = 2 \), \( x = (2)^3 = 8 \).
03
Identify x- and y-intercepts (a)
For \( y = -\sqrt{4-x^2} \), the graph intersects the y-axis where \( x = 0 \), thus the y-intercept is \( y = -2 \). The graph intersects the x-axis where \( y = 0 \), thus the x-intercepts are at \( x = -2 \) and \( x = 2 \).
04
Identify x- and y-intercepts (b)
For \( x = y^3 \), the graph intersects both axes at \( x = 0 \) and \( y = 0 \). Thus, the intercept on both axes is at the origin (0,0).
05
Test for symmetry (a)
The graph of \( y = -\sqrt{4-x^2} \) is symmetric with respect to the y-axis because replacing \( x \) with \( -x \) doesn't change \( y \). This line of symmetry indicates it reflects equally over the y-axis.
06
Test for symmetry (b)
The equation \( x = y^3 \) is symmetric about the origin. If we replace \( (x, y) \) with \( (-x, -y) \), it holds as \( -x = (-y)^3 \). This confirms rotational symmetry about the origin.
07
Sketch Graph of Equation (a)
Utilize the calculated points and symmetry along the y-axis to sketch the semicircle below the x-axis for \( y = -\sqrt{4-x^2} \) with endpoints at \( (-2, 0) \) and \( (2, 0) \).
08
Sketch Graph of Equation (b)
Utilize calculated points to plot the cubic curve for \( x = y^3 \), noting it's symmetric about the origin. It passes through the origin and the points \( (-8, -2) \), \( (-1, -1) \), \( (1, 1) \), and \( (8, 2) \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Intercepts
When graphing equations, finding the intercepts is a crucial step in understanding how the graph interacts with the axes.
- The x-intercepts occur where the graph touches or crosses the x-axis. This happens when the value of y is zero. For example, with the equation \( y = -\sqrt{4-x^2} \), set \( y = 0 \) and solve for x. The solutions, \( x = -2 \) and \( x = 2 \), are the x-intercepts.
- The y-intercept is where the graph touches or crosses the y-axis, at which point x is zero. By setting \( x = 0 \) in the equation, we find that \( y = -2 \) is the y-intercept for the same equation.
Symmetry
Understanding symmetry in graphs can greatly simplify sketching them. Symmetry tells us how certain parts of the graph can mirror others.
- Y-axis Symmetry: If replacing x with -x yields the same equation, the graph is symmetric about the y-axis. For example, with \( y = -\sqrt{4-x^2} \), switching x to -x does not change the equation, hence it shows y-axis symmetry.
- Origin Symmetry: If replacing both x and y with -x and -y holds the equation true, then there is symmetry about the origin. The equation \( x = y^3 \), when tested, retains equality upon flipping both variables, confirming its symmetry about the origin.
Table of Values
Creating a table of values is a fundamental method of plotting graphs. It involves selecting values for one variable and calculating the corresponding values of the other.
- Begin by choosing a set of values for x (or y, depending on the equation). For \( y = -\sqrt{4-x^2} \), x values could be \( -2, -1, 0, 1, 2 \).
- Calculate corresponding y-values by substituting each x value into the equation. This helps visualize the shape and key points of the graph. For example, for \( y = -\sqrt{4-x^2} \), it results in points like \( (-2, 0) \) and \( (0, -2) \).
- For \( x = y^3 \), use values for y to find x. Values such as \( -2, -1, 0, 1, 2 \) produce corresponding x values.
Sketching Graphs
Once you have the intercepts, symmetry, and a table of values, sketching the graph becomes more straightforward.
- Start by plotting the x- and y-intercepts on a coordinate plane. This marks the key points where the graph interacts with the axes.
- Consider the symmetry properties to reflect parts of the graph across axes or the origin. For \( y = -\sqrt{4-x^2} \), knowing it's a semicircle below the x-axis can use the reflected points across the y-axis for simplicity.
- Use the table of values as additional guideposts. Plotting these points gives a clearer picture and continuity to the graph shape. Trace the curve smoothly between points based on the known mathematical relationship.
