Relations and Direct Variation
Consider R = {(-6, 4), (-2, -1), (0, 4), (3, 7), (5, -1)}. State the domain and range. Determine whether R is a function.
Domain = {-6, -2, 0, 3, 5}; Range = {-1, 4, 7}. R is a function,
Rewrite 7x - 3y = 25 in slope-intercept form. State the slope and both intercepts.
y = (7/3)x - 25/3
; slope = 7/3; x-intercept = (25/7, 0); y-intercept = (0, -25/3).
Let f(x) = 2x - 5 and g(x) = f(x + 3) - 4. Describe the transformations and simplify g(x).
Shift left 3 units and down 4 units; g(x) = 2x - 3.
Rewrite
5x - 2y >= 14
in slope-intercept form. State the boundary type and the direction of shading.
y <= (5/2)x - 7
; solid boundary; shade below the line.
A gym charges a registration fee plus a fixed amount per class. Four classes cost $86, and eleven classes cost $156. Write a function for the total cost C(n) and find the cost of eighteen classes.
C(n) = 10n + 46; eighteen classes cost $226.
A direct variation contains the point (-12, 18). Find the constant of variation, write the function rule, and find f(14).
k = -3/2; f(x) = (-3/2)x; f(14) = -21.
A line passes through (-3, 8) and (5, -4). Write its equation in point-slope form and standard form.
y - 8 = (-3/2)(x + 3)
; 3x + 2y = 7.
Let f(x) = -3x + 2 and g(x) = 2f(x - 4) + 1. Describe the transformations and simplify g(x).
Shift right 4 units, stretch vertically by a factor of 2, and shift up 1 unit; g(x) = -6x + 29
Which of these points satisfy 3x + 2y < 5: (-2, -1), (3, -3), (0, 3), and (4, -4)?
(-2, -1), (3, -3), and (4, -4) or all except (0,3)
A tank initially contains 1,250 liters of water. After fourteen minutes of draining at a constant rate, 844 liters remain. Write a function for the remaining volume and determine when the tank will be empty.
V(t) = 1250 - 29t; the tank will be empty after 1250/29 minutes, or approximately 43.1 minutes.
The points (-8, 14), (6, -10.5), and (p, 28) belong to the same direct variation. Find p
p = -16
Write the slope-intercept equation of the line passing through (-3, 5) and perpendicular to 4x + 6y = 9.
y = (3/2)x + 19/2.
The values of f(x) are -4, 0, 6, and 10. Find the corresponding values of h(x) if
h(x) = (-3/2)f(x) + 5.
11, 5, -4, -10.
A dashed boundary line passes through (-2, 5) and (4, -1). The solution region contains the origin. Write the inequality.
y < -x + 3.
A school begins with 672 flyers. After twenty-eight school days, none remain. Write a linear function and interpret the slope.
F(d) = 672 - 24d; the number of flyers decreases by 24 each school day.
For what values of a is
R = {(-5, 2), (a, 4), (3, 2),
(a + 2, 7)} a function?
All real values except a = -7, -5, 1, and 3.
Find the equation in standard form of the line parallel to 3x - 2y = 7 and passing through the intersection of y = 2x + 1 and
x + y = 10.
3x - 2y = -5.
Let f(x) = -0.5x + 5. Compare g(x) = 4[f(x) - 3] and h(x) = 4f(x) - 3. Write simplified rules and describe the difference between their graphs.
g(x) = -2x + 8; h(x) = -2x + 17. The graph of h is 9 units above the graph of g.
For
4x - 3y <= 18
, rewrite in slope-intercept form, describe the line (dashed or solid), and describe the shading.
y >= (4/3)x - 6
; Solid, shaded above.
A festival pass contains 92 credits. Each laser-tag game costs 7 credits, and each climbing session costs 11 credits. A student must complete at least two climbing sessions. Write your inequalities and determine the greatest possible number of laser-tag games.
7x + 11y ≤ 92, y ≥ 2, maximum of 10 laser-tag games.
A direct variation f(x) = kx passes through (a, 18), and f(a + 4) = 30. Find a and k.
a = 6; k = 3
Find the equation in standard form of the line passing through the intersection of 2x - 3y = 7 and x + 2y = 8, and perpendicular to
5x + y = 4.
x - 5y = -1.
Let f(x) = mx + b. The transformation
g(x) = -2f(x - 3) + 5 produces g(x) = 8x - 19. Find m and b.
m = -4; b = 0.
A dashed boundary line is perpendicular to 2x - 5y = 10 and passes through (4, -3). The solution region contains the origin. Write the inequality.
y < (-5/2)x + 7
or 5x + 2y < 14.
Kekos printing charges $180 plus $0.08 per page. Fedex charges $75 plus $0.15 per page. For what number of page totals is Kekos cheaper than Fedex?
At least 1,500 pages. The plans both cost $300 at 1,500 pages, and Kekos is cheaper for more than 1,500 pages.