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mathematics
precalculus

Questions and Answers of Precalculus

On the screen you will see the graph of the unit circle with a point C labeled. Use your mouse and move point C around the unit circle in the counterclockwise direction.What do you notice? In
Find an application in your major field that leads to a sinusoidal graph. Write a paper about your findings.
Explain how the amplitude and period of a sinusoidal graph are used to establish the scale on each coordinate axis.
Explain the term period as it relates to the graph of a sinusoidal function.
Explain the term amplitude as it relates to the graph of a sinusoidal function.
Explain how you would scale the x-axis and y-axis before graphing y = 3 cos(πx).
Find an application in your major field that leads to a sinusoidal graph. Write a paper about your findings.
Explain how the amplitude and period of a sinusoidal graph are used to establish the scale on each coordinate axis.
Is there any function of the form y = 0 < a < 1, that increases more slowly than a logarithmic function whose base is greater than 1? Explain.
Do you think that there is a power function that increases more rapidly than an exponential function whose base is greater than 1? Explain.
Explain in your own words what the number e is. Provide at least two applications that use this number.
Draw the graph of a one-to-one function that contains the points (-2, -3), (0, 0), and (1, 5). Now draw the graph of its inverse. Compare your graph to those of other students. Discuss any
You have just contracted to buy a house and will seek financing in the amount of $100,000.You go to several banks. Bank 1 will lend you $100,000 at the rate of 8.75% amortized over 30 years with a
Explain in your own words the meaning of present value.
Explain in your own words what the term compound interest means.What does continuous compounding mean?
Write a few paragraphs that provide a general strategy for graphing a polynomial function. Be sure to mention the following: degree, intercepts, end behavior, and turning points.
Design a rational function with the following characteristics: three real zeros, one of multiplicity 2; y-intercept 1; vertical asymptotes x = -2 and x = 3; oblique asymptote y = 2x + 1. Is this
Design a polynomial function with the following characteristics: degree 6; four real zeros, one of multiplicity 3; y-intercept 3; behaves like y = -5x6 for large values of |x|. Is this polynomial
Explain the circumstances under which the graph of a rational function will have a hole.
Create a rational function with the following characteristics: three real zeros, one of multiplicity 2; y-intercept 1; vertical asymptotes, x = -2 and x = 3; oblique asymptote, y = 2x + 1. Is this
Write a few paragraphs that provide a general strategy for graphing a rational function. Be sure to mention the following: proper, improper, intercepts, and asymptotes.
Make up a quadratic function that opens down and has only one x-intercept. Compare yours with others in the class. What are the similarities? What are the differences?
Investigate when, historically, the use of the function notation y = f(x) first appeared.
Open the reflection about the x-axis applet. Move your mouse to grab the slide and change the value of a from 1 to -1.
Open the reflection about the y-axis applet. Move your mouse to grab the slide and change the value of a from 1 to -1.
Horizontal Stretches Open the horizontal stretch apple.(a) Use your mouse to grab the slider and change the value of a. Note the role a plays in the graph of g(x) = f(ax) = √ax, where f(x) = √x.
Vertical Stretches Open the vertical stretch applet. Use your mouse to grab the slider and change the value of a. Note the role a plays in the graph of g(x) = af(x), where f(x) = |x|.
Horizontal Shifts Open the horizontal shift applet Use your mouse to grab the slider and change the value of h. Note the role h plays in the graph of g(x) = f(x - h), where f(x) = x2.
Open the vertical shift applet. Use your mouse to grab the slider and change the value of k. Note the role k plays in the graph of g(x) = f(x) + k, where f(x) = x2.
Open the slope applet. Move point B around the Cartesian plane with your mouse.(a) Move B to the point whose coordinates are (2, 7). What is the slope of the line?(b) Move B to the point whose
Which form of the equation of a line do you prefer to use? Justify your position with an example that shows that your choice is better than another. Have reasons.
Carpenters use the term pitch to describe the steepness of staircases and roofs. How does pitch relate to slope? Investigate typical pitches used for stairs and for roofs. Write a brief essay on your
The term grade is used to describe the inclination of a road. How does this term relate to the notion of slope of a line? Is a 4% grade very steep? Investigate the grades of some mountainous roads
The accepted symbol used to denote the slope of a line is the letterm.Investigate the origin of this symbolism. Begin by consulting a French dictionary and looking up the French word monter.Write a
Open the origin symmetry applet. Move point A around the Cartesian plane with your mouse. How are the coordinates of point A and the coordinates of point B related?
Open the x-axis symmetry applet. Move point A around the Cartesian plane with your mouse. How are the coordinates of point A and the coordinates of point B related?
Open the y-axis symmetry applet. Move point A around the Cartesian plane with your mouse. How are the coordinates of point A and the coordinates of point B related?
Draw a graph that contains the points (-2, -1), (0, 1), (1, 3) and (3, 5). Compare your graph with those of other students. Are most of the graphs almost straight lines? How many are “curved”?
Make up an equation with the intercepts (2, 0), (4, 0), and (0, 1). Compare your equation with a friend’s equation. Comment on any similarities.
Write a paragraph that describes a Cartesian plane. Then write a second paragraph that describes how to plot points in the Cartesian plane.Your paragraphs should include the terms “coordinate
Open the “Circle: the role of the radius” applet. Place the cursor on point B, press and hold the mouse button. Drag B around the Cartesian plane.(a) What is the center of the circle?(b) Move B
Open the “Circle: the role of the center” applet. Place the cursor on the center of the circle and hold the mouse button. Drag the center around the Cartesian plane and note how the equation of
Explain how the center and radius of a circle can be used to graph the circle.
Create four problems that you might be asked to do given the two points (-3, 4) and (6, 1). Each problem should involve a different concept. Be sure that your directions are clearly stated.
A car leaves Detroit at 2:00 pm, traveling at a constant speed west along I-96. It passes Ann Arbor, 40 mi from Detroit, at 2:50 pm.(a) Express the distance traveled in terms of the time elapsed.(b)
(a) Show that if the x- and y-intercepts of a line are nonzero numbers a and b, then the equation of the line can be put in the form x/a + y/b = 1This equation is called the two-intercept form of an
(a) Find equations for the sides of the triangle with vertices P(1, 0), Q(3, 4), and R(-1, 6).(b) Find equations for the medians of this triangle. Where do they intersect?
Find an equation of the perpendicular bisector of the line segment joining the points A(1, 4) and B(7, -2).
Show that the lines 3x - 5y + 19 = 0 and 10x + 6y - 50 = 0 are perpendicular and find their point of intersection.
Show that the lines 2x - y = 4 and 6x - 2y = 10 are not parallel and find their point of intersection.
Find the lengths of the medians of the triangle with vertices A(1, 0), B(3, 6), and C(8, 2). (A median is a line segment from a vertex to the midpoint of the opposite side.)
Find the midpoint of the line segment joining the given points.(a) (1, 3) and (7, 15) (b) (-1, 6) and (8, -12)
Show that the midpoint of the line segment from P1(x1, y1) to P2(x2, y2) is (x1 + x2/2, y1 + y2/2)
Find a point on the y-axis that is equidistant from (5, -5) and (1, 1).
Sketch the region in the xy-plane. {(x, y)|2x < y < 1/2 (x + 3)}
Sketch the region in the xy-plane. {(x, y) |1 + x < y < 1 - 2x}
Sketch the region in the xy-plane. {(x, y) | y > 2x - 1}
Sketch the region in the xy-plane. {(x, y) | 0 < y < 4 and x < 2}
Sketch the region in the xy-plane. h(x, y)||x| < 3 and |y| < 2}
Sketch the region in the xy-plane. {(x, y) ||x | < 2}
Sketch the region in the xy-plane. {(x, y) | x > 1 and y < 3}
Sketch the region in the xy-pSketch the region in the xy-plane. lane.{(x, y) | xy < 0}
Sketch the region in the xy-plane. Sketch the region in the xy-plane. {(x, y) | y > 0}
Sketch the region in the xy-plane. {(x, y) | x < 0}
Find the slope and y-intercept of the line and draw its graph.4x + 5y = 10
Find the slope and y-intercept of the line and draw its graph.3x - 4y = 12
Find the slope and y-intercept of the line and draw its graph.2x - 3y + 6 = 0
Find the slope and y-intercept of the line and draw its graph.y = -2
Find the slope and y-intercept of the line and draw its graph.2x - 5y = 0
Find the slope and y-intercept of the line and draw its graph.x + 3y = 0
Find an equation of the line that satisfies the given conditions.Through (1/2 , -2/3), perpendicular to the line 4x - 8y = 1
Find an equation of the line that satisfies the given conditions.Through (-1, -2), perpendicular to the line 2x + 5y + 8 = 0
Find an equation of the line that satisfies the given conditions.y-intercept 6, parallel to the line 2x + 3y + 4 = 0
Find an equation of the line that satisfies the given conditions.Through (1, -6), parallel to the line x + 2y = 6
Find an equation of the line that satisfies the given conditions.Through (4, 5), parallel to the y-axis
Find an equation of the line that satisfies the given conditions.Through (4, 5), parallel to the x-axis
Find an equation of the line that satisfies the given conditions.x-intercept -8, y-intercept 6
Find an equation of the line that satisfies the given conditions.x-intercept 1, y-intercept -3
Find an equation of the line that satisfies the given conditions.Slope 2/5, y-intercept 4
Find an equation of the line that satisfies the given conditions.Slope 3, y-intercept -2
Find an equation of the line that satisfies the given conditions.Through (-1, -2) and (4, 3)
Find an equation of the line that satisfies the given conditions.Through (2, 1) and (1, 6)
Find an equation of the line that satisfies the given conditions.Through (-3, -5), slope -7/2
Find an equation of the line that satisfies the given conditions.Through (1, 7), slope 2/3
Find an equation of the line that satisfies the given conditions.Through (2, -3), slope 6
Sketch the graph of the equation.|y| = 1
Sketch the graph of the equation.y = -2
Sketch the graph of the equation..xy = 0 Find an equation of the line that satisfies the given conditions.
Sketch the graph of the equation.x = 3
Show that A(1, 1), B(11, 3), C(10, 8), and D(0, 6) are vertices of a rectangle.Sketch the graph of the equation.
Show that A(1, 1), B(7, 4), C(5, 10), and D(-1, 7) are vertices of a parallelogram.
(a) Show that the points A(-1, 3), B(3, 11), and C(5, 15) are collinear (lie on the same line) by showing that |AB | + |BC | = |AC|.(b) Use slopes to show that A, B, and C are collinear.
Show that the points (-2, 9), (4, 6), (1, 0), and (-5, 3) are the vertices of a square.
(a) Show that the triangle with vertices A(6, 27), B(11, -3), and C(2, -2) is a right triangle using the converse of the Pythagorean Theorem.(b) Use slopes to show that ABC is a right triangle.(c)
Show that the triangle with vertices A(0, 2), B(-3, -1), and C(-4, 3) is isosceles.
Find the slope of the line through P and Q.P(-1, -4), Q(6, 0)
Find the slope of the line through P and Q.P(-3, 3), Q(-1, -6)
Find the slope of the line through P and Q.P(-1, 6), Q(4, -3)
Find the slope of the line through P and Q.P(1, 5), Q(4, 11)
Find the distance between the points.(a, b), (b, a)

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