Sketch the graph of the equation. x + 2y + 6 = 0
> Prove that if the points (x1, y1) and (x2, y2) lie on the same line as (x1∗, y1∗) and (x2∗, y2∗), then Assume x1 ≠x2 and x1∗ â‰
> Sketch the graph of the equation by point plotting. y = │x + 1│
> Prove that the figure formed by connecting consecutive midpoints of the sides of any quadrilateral is a parallelogram.
> Prove that the diagonals of a rhombus intersect at right angles. (A rhombus is a quadrilateral with sides of equal lengths.)
> Use the result below to find the distance between the point and line. Point: (2, 3) Line: 4x + 3y = 10
> Use the result below to find the distance between the point and line. Point: (−2, 1) Line: x − y − 2 = 0
> Write the distance d between the point (3, 1) and the line y = mx + 4 in terms of m. Use a graphing utility to graph the equation. When is the distance 0? Explain the result geometrically.
> Show that the distance between the point (x1, y1) and the line Ax + By + C = 0 is Distance = /
> An instructor gives regular 20-point quizzes and 100-point exams in a mathematics course. Average scores for six students, given as ordered pairs (x, y), where x is the average quiz score and y is the average exam score, are (18, 87), (10, 55), (19, 96),
> A real estate office manages an apartment complex with 50 units. When the rent is $780 per month, all 50 units are occupied. However, when the rent is $825, the average number of occupied units drops to 47. Assume that the relationship between the monthl
> Find the domain of the function. f(x) = √x2 − 3x + 2
> As a salesperson, you receive a monthly salary of $2000, plus a commission of 7% of sales. You are offered a new job at $2300 per month, plus a commission of 5% of sales. Write linear equations for your monthly wage W in terms of your monthly sales s for
> Sketch the graph of the equation by point plotting. y = (x − 3)2
> Find a linear equation that expresses the relationship between the temperature in degrees Celsius C and degrees Fahrenheit F. Use the fact that water freezes at 0°C (32°F) and boils at 100°C (212°F). Use the equation to convert 72°F to degrees Celsius.
> Several lines are shown in the figure below. (The lines are labeled a–f.) a. Which lines have a positive slope? b. Which lines have a negative slope? c. Which lines appear parallel? d. Which lines appear perpendicular?
> Find the coordinates of the point of intersection of the given segments. Explain your reasoning. a. Perpendicular bisectors b. Medians
> Find an equation of the line tangent to the circle (x − 1)2 + (y − 1)2 = 25 at the point (4, −3).
> Find an equation of the line tangent to the circle x2 + y2 = 169 at the point (5, 12).
> A line is represented by the equation ax + by = 4. When is the line parallel to the x-axis? When is the line parallel to the y-axis? Give values for a and b such that the line has a slope of 5/8. Give values for a and b such that the line is perpendicula
> Show that the points (−1, 0), (3, 0), (1, 2), and (1, −2) are vertices of a square.
> Determine whether the points are collinear. (Three points are collinear if they lie on the same line.) (0, 4), (7, -6), (-5, 11)
> Determine whether the points are collinear. (Three points are collinear if they lie on the same line.) (-2, 1), (-1, 0), (2, -2)
> Find the domain of the function. f(x) = √x + √1 − x
> You are given the dollar value of a product in 2016 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value V of the product in terms of the year t. (Let t = 0 repr
> You are given the dollar value of a product in 2016 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value V of the product in terms of the year t. (Let t = 0 repr
> Write the general forms of the equations of the lines that pass through the point and are a. parallel to the given line and b. perpendicular to the given line. Point: (5/6, - (1/2)) Line: 7x + 4y = 8
> Write the general forms of the equations of the lines that pass through the point and are a. parallel to the given line and b. perpendicular to the given line. Point: (3/4, 7/8) Line: 5x - 3y = 0
> Write the general forms of the equations of the lines that pass through the point and are a. parallel to the given line and b. perpendicular to the given line. Point: (2, 5) Line: x - y = -2
> Write the general forms of the equations of the lines that pass through the point and are a. parallel to the given line and b. perpendicular to the given line. Point: (-3, 2) Line: x + y = 7
> Write the general forms of the equations of the lines that pass through the point and are parallel to the given line and perpendicular to the given line. Point: (-1, 0) Line: y = -3
> Write the general forms of the equations of the lines that pass through the point and are parallel to the given line and perpendicular to the given line. Point: (-7, -2) Line: x = 1
> Use the below result to write an equation of the line with the given characteristics in general form. Point on line: (-(2/3), -2) x-intercept: (a, 0) y-intercept: (0, a) (a ≠0)
> Use the below result to write an equation of the line with the given characteristics in general form. Point on line: (9, -2) x-intercept: (2a, 0) y-intercept: (0, a) (a ≠0)
> Sketch the graph of the equation by point plotting. y = 4 − x2
> Find the domain and range of the function. f(x) = x – 2 / x+4
> Use the below result to write an equation of the line with the given characteristics in general form. x-intercept: (-2/3, 0) y-intercept: (0, -2)
> Use the below result to write an equation of the line with the given characteristics in general form. x-intercept: (2, 0) y-intercept: (0, 3)
> Show that the line with intercepts (a, 0) and (0, b) has the following equation. x/a + y/b = 1, a ≠ 0, b ≠ 0
> Write an equation for the line that passes through the points (0, b) and (3, 1).
> Find an equation of the line that passes through the points. Then sketch the line. (2, 5), (2, 7)
> Find an equation of the line that passes through the points. Then sketch the line. (3, 1), (5, 1)
> Find an equation of the line that passes through the points. Then sketch the line. (1, -2), (3, -2)
> Find an equation of the line that passes through the points. Then sketch the line. (6, 3), (6, 8)
> Find an equation of the line that passes through the points. Then sketch the line. (-3, 6), (1, 2)
> Sketch the graph of the equation by point plotting. y = 5 − 2x
> Find an equation of the line that passes through the points. Then sketch the line. (2, 8), (5, 0)
> Find the domain and range of the function. f(x) = 3/x
> Find an equation of the line that passes through the points. Then sketch the line. (-2, -2), (1, 7)
> Find an equation of the line that passes through the points. Then sketch the line. (4, 3), (0, -5)
> Sketch the graph of the equation. 3x – 3y + 1 = 0
> Sketch the graph of the equation. y – 1 = 3(x + 4)
> Sketch the graph of the equation. y – 2 = 3/2 (x – 1)
> Sketch the graph of the equation. y = 1 / 3 x - 1
> Sketch the graph of the equation. y = -2x + 1
> Sketch the graph of the equation by point plotting. y = (1/2)x + 2
> Sketch the graph of the equation. x = 4
> Sketch the graph of the equation. y = -3
> Find the domain and range of the function. f(x) = │x − 3│
> Find the slope and the y-intercept (if possible) of the line. y = -1
> Find the slope and the y-intercept (if possible) of the line. x = 4
> Find the slope and the y-intercept (if possible) of the line. 6x - 5y = 15
> Find the slope and the y-intercept (if possible) of the line. 5x + y = 20
> Find the slope and the y-intercept (if possible) of the line. -x + y = 1
> Find the slope and the y-intercept (if possible) of the line. y = 4x – 3
> The table shows the biodiesel productions y (in thousands of barrels per day) for the United States for 2007 through 2012. The variable t represents the time in years, with t = 7 corresponding to 2007. (Source: U.S. Energy Information Administration) a.
> Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] y = x3 – x
> The table shows the populations y (in millions) of the United States for 2009 through 2014. The variable t represents the time in years, with t = 9 corresponding to 2009. (Source: U.S. Census Bureau) a. Plot the data by hand and connect adjacent points
> A moving conveyor is built to rise 1 meter for each 3 meters of horizontal change. Find the slope of the conveyor. Suppose the conveyor runs between two floors in a factory. Find the length of the conveyor when the vertical distance between floors is 10
> You are driving on a road that has a 6% uphill grade. This means that the slope of the road is 6 100. Approximate the amount of vertical change in your position when you drive 200 feet.
> Find the domain and range of the function. f(x) = √16 − x2
> Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point: (-2, 4) Slope: m = -(3/5)
> Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point: (3, -2) Slope: m = 3
> Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point: (0, 4) Slope: m = 0
> Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point: (1, 2) Slope: m is undefined.
> Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point: (-5, -2) Slope: m = 6/5
> Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point: (0, 3) Slope: m = 3/4
> Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] y = 3 − x2
> Use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) Point: (-2, -1) Slope: m = 2
> Use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) Point: (1, 7) Slope: m = -3
> Use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) Point: (-4, 3) Slope: m is undefined
> Use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) Point: (6, 2) Slope: m = 2
> Find the domain and range of the function. h(x) = −√x + 3
> Consider a polynomial f(x) with real coefficients having the property f(g(x)) = g( f(x)) for every polynomial g(x) with real coefficients. Determine and prove the nature of f(x).
> Sketch the lines through the point with the indicated slopes. Make the sketches on the same set of coordinate axes. Point (-2, 5) Slopes a. 3 b. -3 c. 1/3
> Plot the pair of points and find the slope of the line passing through them. (7/8, 3/4), (5/4, - (1/4)
> Plot the pair of points and find the slope of the line passing through them. (-(1/2, 2/3), (-(3/4), 1/6
> Plot the pair of points and find the slope of the line passing through them. (3, −5), (5, −5)
> Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] y = √9 – x2
> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f(x) = f(−x) for all x in the domain of f, then the graph of f is symmetric with respect to the y-axis.
> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. A vertical line can intersect the graph of a function at most once.
> Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f(a) = f(b), then a = b
> 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 squ
> A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point (3, 2) (see figure). Write the length L of the hypotenuse as a function of x
> Prove that the product of an odd function and an even function is odd.
> Prove that the product of two even (or two odd) functions is even.
> Evaluate the function at the given value(s) of the independent variable. Then find the domain and range. a. f (−3) b. f (0) c. f (5) d. f (10)
> Sketch the lines through the point with the indicated slopes. Make the sketches on the same set of coordinate axes. Point (3, 4) Slopes 1 -2 -(3/2)
> Find any intercepts. y2 = x3 − 4x
> Find any intercepts. y = x2 + x – 2
> A skydiver, who weighs 650 N, is falling at a constant speed with his parachute open. Consider the apparatus that connects the parachute to the skydiver to be part of the parachute. The parachute pulls upward on the skydiver with a force of 620 N. (a) Id
