MATH 115: Review for Chapter 1 Can you use the Distance Formula to find the distance between two points? (1) Find the distance d P, P between the points P and 1 1, 6 P 10,9. () Find the length of the line segment from the points A 7, 4 to,8 B. (3) Find all points having a y-coordinate of -3 whose distance from the point,5 is 10. Can you use the Midpoint Formula to find the midpoint of a line segment? P and (4) Find the midpoint of the line segment joining the points 1 5,4 P. 3, 7 (5) One of the endpoints of a line segment has coordinates 3, 1. The midpoint of the line segment is, 3. Find the coordinates of the other endpoint. Can you solve application problems involving the Distance Formula and the Midpoint Formula? (6) Determine whether the triangle formed by the vertices A 3,, B 0,4,1, and C is an isosceles triangle, a right triangle, neither of these, or both. (7) Determine whether the triangle formed by the midpoints of the three sides of the triangle with vertices 0,0, 4,4 3 is an equilateral triangle. 8,0, and Can your determine the domain and range a relation, and tell whether the relation is a function? Determine the domain and range of each relation, and tell whether the relation is a function. Assume that the graphs extends indefinitely and a table includes only the points shown. (8)(9,1), (6,5), (10,5), (4,1)} (9) x 0 1 0 y 5 3-4 Revised January 016 1
(10) (11) (1) Can you evaluate a function at given points? (13) Given (14) Given 1, evaluate 0 f x x x x 1 f x x 1, evaluate 1 f, f, and f and 1 f. f. Can you graph an equation by hand by plotting points, label the x- and y- intercepts, and verify your results using a graphing utility? (15) Graph each equation by hand by plotting points. Be sure to label the x- and y- intercepts, and verify your results using a graphing utility. (a) 4y 3x 1 y 4 x y x 3 Revised January 016
Given the graph of a function, can you find the intercepts, the value of x given y, and the value of y given x? (16) Use the curve below to answer the following questions. Assume that each box measures 1 unit x 1 unit. (a) What are f 0 and 4 f? Is f 3 positive or negative? What is the domain of f? (d) What are the x- and y- intercepts? (e) For what values of x does f x 1? Can you find the slope of a line given two points on the line? (17) Determine the slope (if it is defined) of each line below containing the given points. (a) 4, ; 3,4 1, ; 1, Can you find an equation of a line given its graph? (18) Find an equation for each line below. (a) Revised January 016 3
Can you find the slope and y-intercept for a given linear equation? (19) Find the slope and the y-intercept for each equation. (a) x y 1 1 1 y x 3x y 6 (d) y 4 0 Can you find a linear equation given specific conditions? (0) Find an equation for each line with the given conditions. Express your answer in slope-intercept form or general form. (a) slope = ; containing the point 4, 3 containing the points 3,4 and,5 parallel to the line y 3x (d) containing the points 4,3 and 4, 3 ; containing the point 1, (e) slope undefined; containing the point,4 (f) slope zero; containing the point,3 Given an equation of a line, can you find the x- and y-intercepts? (1) For each equation below, find both the x- and y-intercepts. (a) x y 4 x 3y 9 Can you graph a linear equation? () Graph each equation below by hand. Include a table of values. (a) 3x y 6 1 3 x y Revised January 016 4
y 3x 4 (d) y Given a set of data relating two variables, and using a graphing utility, can you find the linear regression line? (3) Consider the following data, which relates the distance, d, in miles driven by a car, to the time, t, that the car has been driven. t (hours) d (miles) 0 0 1 45 8 3 15 4 5 5 315 6 390 (a) (d) (e) (f) (g) Draw a scatter diagram using a graphing utility, and then find the line of best fit relating the distance to time. Does the relationship between d and t represent a function? Interpret the slope of the line of best fit. Express the relationship found in part (a) using function notation. What is the domain of the function? Predict the distance the car has traveled after 1 hours. Estimate the distance traveled after 5.75 hours. Revised January 016 5
(4) An ecologist tracked and studied 145 deer that were born in 1987. The number of these deer still leaving each year in the study is shown in the table. Year 1987 1988 1989 1990 1991 199 1993 1994 1995 S 145 144 134 103 70 45 3 4 Let t 0 represent 1987. (a) (d) Make a scatter diagram of the data using your graphing utility. Use your graphing utility to find the line of best fit to the data. Round your numbers to whole integer values. Graph the line of best fit on the scatter diagram using your graphing utility. Interpret the slope of the line of best fit. (e) Predict the number of deer still living in 1996. Can you solve a linear equation algebraically and verify your solution using a graphing utility? (5) Solve each equation algebraically. Verify your solution using a graphing utility. (a) x 5 3 3 6 x 1 x x 3 3 p 4p 5 5 p 3 Can you solve linear inequalities and combined (double) inequalities algebraically and graphically and use interval notation to write the solution? (6) Solve each inequality (a) algebraically and graphically. Use interval notation to write the solution. (a) 5x 7x 6 1 x 4 3x 3 4 m 7 5 0 (d) 3x 5 x 4 (e) 7 5 t t 3 (f) 5 x 3 x 4 3x 19 (g) 5 r 3 1 (h) 1 9 5t 9 (i) x 13 7 3 31 (j) x 1 3 3 4 Revised January 016 6
(k) w 0 3 4 6 Answers: d P, P 17 (1) () 13 1 (3) 4, 3 and 8, 3 (4) 3 4, (5) 7, 5 (6) The triangle is both an isosceles and a right triangle. So it is an isosceles right triangle. (7) Yes, it is an equilateral triangle because each side of this new triangle has length 4. (8) The domain is {4,6,9,10} and the range is {1,5}. The relation is a function since each element in the domain corresponds to exactly one element in the range. (9) The domain is {0,1} and the range is {-4,3,5}. The relation is not a function since the {1} in the domain is assigned two values in the range. (10) domain: all real numbers; range: y y. The relation is a function since the graph passes the vertical line test. (11) domain: x x 1 ; range: 0 y y. The relation is a function since the graph passes the vertical line test. (1) domain: { x x } ; range: { y 3 y 3}. The relation is not a function since the graph fails the vertical line test. (13) f 0 1; f 7; (14) f 1 0; 1 f 11 f is undefined (15) (a) Revised January 016 7
note: The line must pass through the origin O 0,0. (16) (a) 1.5 positive 8,8 (d) x-intercepts: -8, -3.5, -0.75, and 5; y-intercept: 1.5 x 7.5, 4., 0.,3.4 (e) (17) (a) m slope is undefined (18) (a) y x y x (19) (a) m 1 m y-intercept 1 y-intercept 3 m (d) m 0 y-intercept 3 y-intercept 4 (0) (a) y x 11 or x y 11 y 1 3 x or x 5y 3 5 5 y 3x 1 or 3x y 1 (d) x 4 (e) x (f) y 3 (1) (a) x-intercept ; y-intercept 4 x-intercept 9 ; y-intercept 3 Revised January 016 8
() (a) y 4.0 4.0 3.0 3.0.0.0 1.0 1.0 x -4.0-3.0 -.0-1.0 1.0.0 3.0 4.0 5.0-1.0-4.0-3.0 -.0-1.0 1.0.0 3.0 4.0 5.0-1.0 -.0 -.0-3.0-3.0-4.0-4.0 (d) 4.0 4.0 3.0 3.0.0.0 1.0 1.0-4.0-3.0 -.0-1.0 1.0.0 3.0 4.0 5.0-1.0-4.0-3.0 -.0-1.0 1.0.0 3.0 4.0 5.0-1.0 -.0 -.0-3.0-3.0-4.0-4.0 (3) (a) y 66.18 x 9.68 yes distance increases at the rate of approximately 66.18 mph (d) f t 66.18 t 9.68 (e) t 0 (f) approximately 764.48 miles (g) approximately 350.86 miles (4) (a) Let S represent still-living deer and t represent the number of years since 1987. Then S 0t 157. Revised January 016 9
(d) The slope of 0 tells us that, on average, 0 deer die per year. (e) none (5) (a) x 4 7 x p (6) (a) 3 x interval notation: 3, 7 x interval notation: 14 3 m interval notation: 4 7, 14 3, 4 x interval notation:, 9 (d) 9 (e) t 5 interval notation: 5, x interval notation:, 3 (f) 3 (g) r 4 interval notation:,4 interval notation: 4, (h) 4 t interval notation: 9,0 (i) 9 x 0 (j) 5 3 x 5 interval notation: 5,5 3 (k) 0 w interval notation: 0, Revised January 016 10