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Line L passes through the points (1,3) and (-3,4).
a) Fine the slope of line L.
b) Find the y-intercept of L.
c) Find the x-intercept of L.
d) Find b so that line L passes through the point (-4,b).
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The graph of function f is shown below.
a) Find the domain of f.
b) Find the range of f.
c) Find f(0), f(1) and f(2).
d) Find all values of x for which f(x) = 5.
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The graph of line segment AB is shown below.
a) Use the graph to find the coordinates of points A and B.
b) Find the length of the line segment AB.
c) Find the midpoint of the line segment AB.
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Line L1 passes through the point (1 , b) as shown below.
Find the equation of line L2 that passes through point A and is perpendicular to line L1.
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Are the two lines L1 through the points (-1,4) and (0,4) and L2 through the points (5,6) and (5,2) parallel, perpendicular or neither. Explain your answer.
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Find all values of the constant K so that the inequality 5(x - K + 2) ≥ 2(x + 4) - 7 has a solution set given by the interval [10 , + infinity).
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Linda walked for 2 hours then ran for 1 hour. If she runs three times as fast as she walks and the total trip was 20 kilometers, then how fast does she run?
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Each pair of the three lines defined by their equations y = (1/2) x - 1, y = 2x + 2 and y = -x + 2 has a point of intersection so that when put together there are three points of intersection making a triangle ABC.
Show that triangle ABC is isosceles.
Answers to the Above Questions
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a) slope = (4 - 3) / (-3 - 1) = -1/4
b) Equation of line L: y - 3 = (-1/4)(x - 1)
To find the y-intercept, put x = 0 in the equation: y - 3 = 1/4 , y = 13/4
c) To find the x-intercept, put y = 0 in the equation: 0 - 3 = (-1/4)(x - 1) , x = 13
d) Substitute x and y by -4 and b respectively in the equation of line L and solve for b.
b - 3 = (-1/4)(-4 - 1) , b = 17/4
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a) Domain: (-4 , 4]
b) Range: [-4 , 12]
c) f(0) = -4, f(1) = -3, f(2) = 0
d) f(x) = 5 for x = -3 or x = 3
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a) A(-2 , 4) , B(4 , 2)
b) length of AB = √[ (2 - 4) 2 + (2 - 4) 2 ] = 2√10
c) Midpoint coordinates: x = (-2+4)/2 = 1 , y = (4+2)/2 = 3
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From graph, x-intercept (-3 , 0), y-intercept (0 , 2)
Slope of L1 = (2 - 0) / (0 - (-3)) = 2/3
Find b: slope using the y-intercept and point (1 , b)
(b - 2)/(1 - 0) = 2/3
solve to find b = 8/3
Slope of L2 = - 1/slope of L1 = -3/2 (relationship between slopes of perpendicular lines)
equation of L2: y - 8/3 = -3/2 (x - 1) or y = (-3/2) x + 25/6
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Slope of L1 = (4 - 4) / (0 + 1) = 0 , L1 is horizontal
Slope of L2 = (6 - 2) / (5 - 5) = undefined , L2 is vertical
The two lines are perpendicular.
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Solve the given inequality 5(x - K + 2) ≥ 2(x + 4) - 7 to obtain the solution set
x ≥ -3 + 5K/3
For the above set to be equal to the set [10 , +infinity) we need to have
-3 + 5K/3 = 10
Solve the above for K to obtain: K = 39/5
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Let x be the walking speed. 3x will be the running speed. 2*(x) will be the distance walked and 1*(3x) will be the distance run. Total distance is 20 km. Hence
2x + 3x = 20
x = 4 km/hr , 3x = 12 km/hr and it is the speed of Linda when running.
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let A be the point of intersection of the lines y = (1/2) x - 1 and y = 2x + 2. Solve the system of equations made by the equations of the two lines to find the point.
(1/2) x - 1 = 2x + 2 , solve for x to find x = -2 and substitute to find y = -2. Hence point A has coordinates (-2 , -2).
Let point B be the point of intersection of the lines y = (1/2) x - 1 and y = -x + 2. using the same ideas as above to find the coordinates of B as (2 , 0).
Let point C be the point of intersection of the lines y = -x + 2 and y = 2x + 2. And similar ideas can be used to find the coordinates of C as (0 , 2).
length of line segment AB = √(2 2 + 4 2) = 2√5
length of line segment AC = √(4 2 + 2 2) = 2√5
length of line segment BC = √(2 2 + 2 2) = 2√2
length of AB = length of AC and therefore the triangle is isosceles.
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