Tip: In a table, find the column where x = 0 — not the first column and not the largest number.
Question 4 of 36
No calculator
Line g appears on the grid below. Function f has the rule f(x) = -x + 4.
Of these two functions, which one decreases faster? Give its rate of change.
Hint
Find g's slope from the two marked points (rise ÷ run). "Decreases faster" means a steeper downhill line.
Step-by-step solution
f(x) = -x + 4 means f(x) = -1x + 4, so the rate of change of f is -1.
g passes through (0, 1) and (2, -3).
Rate of change of g: m = \dfrac{-3 - 1}{2 - 0} = \dfrac{-4}{2} = -2.
Both are negative, so both decrease. Compare steepness (the size of the slope): g drops 2 per unit, f drops only 1 per unit.
Answer: g, with a rate of change of -2
Check: From x = 0 to x = 2, g falls from 1 to -3 (down 4), while f falls from 4 to 2 (down 2). g falls twice as far ✓
Tip: For decreasing functions, -2 < -1, but -2 is the faster decrease. Compare how far each one drops per step, ignoring the minus sign.
Question 5 of 36
No calculatorFill in
In a science class, students measure two bean plants. The height of Plant A is shown in the table. The height of Plant B, in centimeters, after w weeks is B(w) = 1.5w + 6.
Week w
0
3
6
9
Height of Plant A (cm)
4
10
16
22
How many more centimeters per week does Plant A grow than Plant B?
cm per week
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The weeks in the table go up by 3, not 1. Divide the change in height by the change in weeks.
Step-by-step solution
Plant A's rate: \dfrac{10 - 4}{3 - 0} = \dfrac{6}{3} = 2 cm per week. (Check another pair: \dfrac{22 - 16}{9 - 6} = 2 ✓)
Plant B's rate: in B(w) = 1.5w + 6, the coefficient of w is the rate: 1.5 cm per week. (The 6 is the starting height.)
Difference: 2 - 1.5 = 0.5.
Answer: 0.5 cm per week
Check: Over 6 weeks Plant A grows 16 - 4 = 12 cm; Plant B grows B(6) - B(0) = 15 - 6 = 9 cm. The difference is 3 cm over 6 weeks, or 3 \div 6 = 0.5 cm per week ✓
Tip: Never read a table's change of 6 as "6 per week" without checking the step in the top row. Here 6 cm takes 3 weeks.
Question 6 of 36
No calculator
Two water tanks are draining at steady rates. The amount of water in Tank A is shown in the table. The amount of water in Tank B, in gallons, after t hours is W(t) = 380 - 25t.
Hours t
1
2
3
Water in Tank A (gal)
340
320
300
Which tank held more water at the start (t = 0), and by how much?
Hint
The table does not start at t = 0. Use the hourly change in Tank A to work back one hour.
Step-by-step solution
Tank A loses 340 - 320 = 20 gallons each hour.
Work back from t = 1 to t = 0: add one hour's loss, 340 + 20 = 360. Tank A started with 360 gallons.
Tank B at the start: W(0) = 380 - 25(0) = 380 gallons.
Compare: 380 - 360 = 20.
Answer: Tank B, by 20 gallons
Check: Tank A's rule is A(t) = 360 - 20t: A(3) = 360 - 60 = 300 ✓ matches the table.
Tip: The "start" (the y-intercept) is the value at t = 0. If the table skips 0, extend the pattern backwards one step.
Question 7 of 36
The function f is defined by f(x) = -x^2 + 6x - 4. The graph of the function g is shown.
Which function has the greater maximum value?
Hint
Find the vertex of f with x = -\frac{b}{2a}, then compute f at that x. Read g's highest point from the graph.
Step-by-step solution
f(x) = -x^2 + 6x - 4 has a = -1, b = 6. Since a < 0, it opens down and has a maximum.
Vertex x-coordinate: x = -\frac{6}{2(-1)} = 3.
Maximum value of f: f(3) = -(3)^2 + 6(3) - 4 = -9 + 18 - 4 = 5.
From the graph, g's highest point is (2, 6), so g's maximum value is 6.
Compare: 6 > 5.
Answer: g(x); its maximum value, 6, is greater than f's maximum value, 5.
Check:f(2) = -4 + 12 - 4 = 4 and f(4) = -16 + 24 - 4 = 4 — equal, confirming the vertex of f is at x = 3 ✓
Tip: "Maximum value" means the y-coordinate of the vertex. The x-coordinate only tells you where it happens.
Question 8 of 36
Fill in
The graph shows the balance in Account A over time. Account B starts with $500 and the owner adds $30 each month. Neither account earns interest.
After how many months will the two accounts have the same balance?
months
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find Account A's starting balance and monthly increase from the graph. Write both as equations and set them equal.
Step-by-step solution
Account A from the graph: starts at $200 (the point (0, 200)); rate = \frac{400 - 200}{4 - 0} = \frac{200}{4} = \text{\textdollar}50 per month. So A(m) = 200 + 50m.
g: the outputs are symmetric about x = 0 (g(-1) = g(1) = 0, g(-2) = g(2) = 3), so the smallest output is g(0) = -1.
Compare: -1 < 2, so g has the smaller minimum value.
Answer: g's minimum value, -1, is less than f's minimum value, 2.
Check:f(2) = 4 - 12 + 11 = 3 and f(4) = 16 - 24 + 11 = 3, equal on both sides of x = 3, so 3 is the axis of symmetry and f(3) = 2 is the minimum ✓
Tip: The minimum value is an output (y-value). In a table, find the turning point, where the outputs stop going down and start going up.
Question 14 of 36
Fill in
Two cars set out on a road trip. The fuel left in Car A's tank is shown in the table. The fuel left in Car B's tank, in gallons, after m miles is G(m) = 18 - 0.06m.
Miles driven m
0
50
100
Fuel in Car A (gal)
14
12
10
After how many miles will the two cars have the same amount of fuel left?
miles
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Write a rule for Car A: starting fuel, minus (gallons used per mile) × m. Then set the two rules equal.
Step-by-step solution
Car A's rate: \dfrac{12 - 14}{50 - 0} = \dfrac{-2}{50} = -0.04 gallon per mile. Start: 14 gallons. So A(m) = 14 - 0.04m.
Tip: Car B starts with 4 more gallons but uses 0.02 gallon more per mile. The gap closes at 0.02 per mile: 4 \div 0.02 = 200. That shortcut is a great check.
Question 15 of 36
A family is renting a moving truck for a 120-mile move.
Truck A costs C(m) = 29.95 + 0.79m dollars for m miles.
Truck B charges a flat fee plus a fixed amount per mile, as shown in the table.
Miles
50
100
150
Truck B cost ($)
64.50
94.50
124.50
Which truck is cheaper for this move, and by how much?
Hint
Find Truck B's cost per mile from the table (change in cost ÷ change in miles), then its flat fee. Evaluate both at 120 miles.
Step-by-step solution
Truck B's rate: \dfrac{94.50 - 64.50}{100 - 50} = \dfrac{30}{50} = 0.60 dollar per mile.
Truck B's flat fee: work back from 50 miles: 64.50 - 0.60(50) = 64.50 - 30 = 34.50. So B(m) = 34.50 + 0.60m.
Truck A at 120 miles: C(120) = 29.95 + 0.79(120) = 29.95 + 94.80 = 124.75.
Truck B at 120 miles: B(120) = 34.50 + 0.60(120) = 34.50 + 72 = 106.50.
Difference: 124.75 - 106.50 = 18.25.
Answer: Truck B, by $18.25
Check:B(150) = 34.50 + 90 = 124.50 ✓ matches the table, so the rule for Truck B is right.
Tip: A lower flat fee does not mean a cheaper trip. For a long trip, the per-mile rate matters most; always evaluate both options at the actual input.
Question 16 of 36
Select all
Here are two quadratic functions: f(x) = -x^2 + 4x, and k, whose graph appears below.
Which statements about f and k are true? Select all that apply.
Select all that apply.
Hint
For f: find its vertex with x = -\frac{b}{2a}, its y-intercept f(0), and its zeros by factoring out x. Read the same features of k from the graph.
Step-by-step solution
f:a = -1 < 0, so it opens down and has a maximum. Vertex x = -\dfrac{4}{2(-1)} = 2; f(2) = -4 + 8 = 4. Maximum value 4 at x = 2.
y-intercept: f(0) = 0. Zeros: -x^2 + 4x = -x(x - 4) = 0, so x = 0 or x = 4.
k (graph): opens down, highest point (-1, 9): maximum value 9 at x = -1. y-intercept 8. Zeros -4 and 2.
Test each statement:
Both have a maximum value ✓
k's maximum (9) is greater than f's (4) ✓
f's maximum occurs at x = 2, to the right of k's at x = -1 ✓
Same y-intercept? 0 \ne 8 ✗
Both cross the x-axis at x = 2? f(2) = 4, not 0 ✗
f has a minimum value of 0? f opens down, so it has no minimum ✗
Answer: "Both have a maximum," "k's maximum value is greater," and "f's maximum occurs to the right of k's."
Tip: Make a two-column feature table (opens, vertex, y-intercept, zeros) for both functions before reading the statements.
Question 17 of 36
No calculatorDrop-down
The linear function p passes through the points (1, 7) and (5, -5). The linear function q has the value 10 when x = 0 and decreases by 4 each time x increases by 1.
Complete the statements.
The rate of change of p is .
As x increases, p(x) decreases q(x).
At x = 5, the value of p is the value of q.
Choose an answer from each menu.
Hint
Use the slope formula for p. Write q as an equation from the description. Then evaluate both at x = 5.
Step-by-step solution
Slope of p: m = \dfrac{-5 - 7}{5 - 1} = \dfrac{-12}{4} = -3.
q decreases by 4 per unit, so its rate is -4. p drops 3 per unit and q drops 4, so p decreases more slowly thanq.
Equation of q: q(x) = 10 - 4x, so q(5) = 10 - 20 = -10.
p(5) = -5 (it is one of the given points).
Compare: -5 > -10, so p(5) is greater thanq(5).
Answer: rate of p = -3; p decreases more slowly than q; at x = 5, p is greater than q.
Check:p(x) = -3x + 10: p(1) = 7 ✓ and p(5) = -5 ✓. Both lines start at 10, and q falls faster, so q must be lower at x = 5 ✓
Tip: On a number line, -5 is to the right of -10, so -5 is the greater value even though 10 is "bigger" than 5.
Question 18 of 36
No calculatorDrag and drop
Four linear functions are described on the tiles: an equation, a table of points, a description in words, and another equation.
Drag the correct function into each box. One function is not used.
Least rate of change
Greatest rate of change
Greatest y-intercept
Tap a tile, then tap a box to place it.
Hint
Write down the slope and y-intercept of each function first. For the table, use change in y over change in x; the y-intercept is the output at x = 0.
Tip: A negative rate is always less than any positive rate. And the function with the greatest rate (h) has the lowest starting value here: rate and intercept are separate properties.
Question 19 of 36
Two balls are thrown straight up. The height of Ball A is shown on the graph. The height of Ball B, in feet, after t seconds is
h(t) = -16t^2 + 40t + 4
Which ball reaches the greater maximum height, and by how much?
Hint
Read Ball A's highest point from the graph. For Ball B, find the time of the vertex with t = -\frac{b}{2a}, then evaluate h at that time.
Step-by-step solution
Ball A: the highest point on the graph is (1.5, 40), so its maximum height is 40 ft.
Ball B: a = -16, b = 40. Vertex time t = -\dfrac{40}{2(-16)} = \dfrac{40}{32} = 1.25 seconds.
Check:h(1) = -16 + 40 + 4 = 28 and h(1.5) = -36 + 60 + 4 = 28, equal on both sides of t = 1.25, so Ball B's peak is at t = 1.25 and is higher than 28 ft ✓
Tip: Compare the heights (the y-values of the vertices), not the times. The coefficients 40 and 4 in the equation are not heights you can read off directly.
Question 20 of 36
No calculator
The function f is defined by f(x) = x^2 - 5. The function g is shown in the table.
x
2
4
6
g(x)
9
15
21
Which statement correctly compares f(4) and g(4)?
Hint
Compute f(4) from the equation. Read g(4) from the column where x = 4.
Step-by-step solution
f(4) = 4^2 - 5 = 16 - 5 = 11.
In the table, under x = 4: g(4) = 15.
Compare: 15 - 11 = 4, so g(4) is greater by 4.
Answer: g(4) is greater than f(4) by 4.
Check:g rises 6 for every 2 units, so g(x) = 3x + 3 and g(4) = 12 + 3 = 15 ✓. And f(4) = 11 is less than 15 ✓
Tip: To compare two functions at an input, put both values side by side. For a table, find the column with the same input; do not use a different column.
Question 21 of 36
The graph of the function f is shown. The function g is defined by g(x) = x^2 - 4.
Which statement correctly compares the greatest x-intercept of each function?
Hint
Read where the graph of f crosses the x-axis. For g, solve x^2 - 4 = 0.
Step-by-step solution
x-intercepts are where the output is 0.
f (graph): the curve crosses the x-axis at x = -1 and x = 5. Greatest: 5.
g: solve x^2 - 4 = 0 → x^2 = 4 → x = 2 or x = -2. Greatest: 2.
Compare: 5 > 2.
Answer: f: 5 and g: 2, so f has the greater x-intercept.
Check:g(2) = 4 - 4 = 0 ✓. The graph of f fits f(x) = x^2 - 4x - 5 = (x + 1)(x - 5), and f(5) = 0 ✓
Tip: Don't confuse the three key points of a parabola: the x-intercepts (output 0), the y-intercept (input 0) and the vertex (turning point). The graph of f shows all three.
Question 22 of 36
Fill in
The function f is defined by f(x) = 2.5x - 1. The graph of the linear function h is shown.
What is the rate of change of fminus the rate of change of h?
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find h's slope from the two marked points; it is negative. Subtracting a negative number means adding.
Step-by-step solution
Rate of change of f: the coefficient of x in f(x) = 2.5x - 1 is 2.5.
h passes through (-2, 7) and (4, -2).
Rate of change of h: m = \dfrac{-2 - 7}{4 - (-2)} = \dfrac{-9}{6} = -1.5.
Tip: Job A has the higher base, but Job B's higher rate wins once sales are large enough. Never compare just one feature; evaluate both functions at the actual input.
Question 28 of 36
Select all
The quadratic function f is shown in the table. The function g is defined by g(x) = -2x^2 + 8x - 6.
x
0
1
2
3
4
5
f(x)
5
0
-3
-4
-3
0
Select all true statements.
Select all that apply.
Hint
In the table, find the turning point and where f(x) = 0. For g, find the vertex with x = -\frac{b}{2a}, its y-intercept, and its zeros.
Step-by-step solution
f (table): outputs fall then rise, symmetric about x = 3, so f has a minimum of -4 at x = 3. Zeros: x = 1 and x = 5. y-intercept: 5.
Tip: "Minimum value" is the output at the vertex; "where the minimum occurs" is the input. Read each statement for which one it means.
Question 29 of 36
Two drivers track the fuel in their cars on a highway trip.
Car X had 11.2 gallons left after 30 miles and 9.4 gallons left after 90 miles.
Car Y uses 1 gallon of fuel for every 25 miles it drives.
Which car uses fuel at a faster rate per mile?
Hint
Find Car X's change in fuel per mile from the two data points. Turn "1 gallon every 25 miles" into gallons per mile.
Step-by-step solution
Car X: rate = \dfrac{9.4 - 11.2}{90 - 30} = \dfrac{-1.8}{60} = -0.03 gallon per mile, so it uses 0.03 gallon per mile.
Car Y: 1 gallon per 25 miles = \dfrac{1}{25} = 0.04 gallon per mile.
Compare the amounts used: 0.04 > 0.03, so Car Y uses fuel faster.
Answer: Car Y; it uses 0.04 gallon per mile, compared with 0.03 for Car X.
Check: In 60 miles, Car X used 1.8 gallons; Car Y would use 60 \times 0.04 = 2.4 gallons. Car Y uses more ✓
Tip: "Miles per gallon" and "gallons per mile" are opposites (reciprocals). A car that gets more miles per gallon uses less fuel per mile. Decide which rate the question wants before comparing.
Question 30 of 36
Drag and drop
Four functions are given in different forms on the tiles.
Drag the functions into the boxes to order them from leasty-intercept to greatesty-intercept. Every function is used once.
Least
Greatest
Tap a tile, then tap a box to place it.
Hint
The y-intercept is the output when x = 0. For the table and the two points, find the slope first, then work back to x = 0.
Step-by-step solution
a(x) = 2(x - 3):a(0) = 2(-3) = -6.
b (table): slope \dfrac{7 - 4}{2 - 1} = 3. Work back from x = 1 to x = 0: 4 - 3 = 1. y-intercept 1.
c (line through (2, 1) and (4, -3)): slope \dfrac{-3 - 1}{4 - 2} = -2. Work back 2 units from x = 2: 1 - (-2)(2) = 1 + 4 = 5. y-intercept 5.
Tip: For a(x) = 2(x - 3), the y-intercept is not -3: you must multiply by 2. Always compute f(0) instead of guessing from the numbers you see.
Question 31 of 36
Fill in
Two companies model their weekly profit, in thousands of dollars, from making x hundred units. A company makes a profit only between its two break-even points (the x-values where profit is 0).
Company A:P(x) = -x^2 + 14x - 40
Company B: shown in the table
x (hundreds of units)
1
2
3
6
7
8
Profit (thousands of $)
0
6
10
10
6
0
How many hundred units wider is Company B's profitable range than Company A's?
hundred units
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find Company A's break-even points by solving -x^2 + 14x - 40 = 0 (factor). Read Company B's from the table. Width = larger zero − smaller zero.
Step-by-step solution
Company A: solve -x^2 + 14x - 40 = 0. Multiply by -1: x^2 - 14x + 40 = 0. Factor: (x - 4)(x - 10) = 0, so x = 4 or x = 10.
Width of A's profitable range: 10 - 4 = 6.
Company B: the table shows profit 0 at x = 1 and x = 8.
Tip: Break-even points are the zeros (x-intercepts). The width between them is the larger zero minus the smaller one. Don't confuse them with the x-values of the highest profit (3 and 6 in the table).
Question 32 of 36
No calculatorDrop-down
The function f is defined by f(x) = -x^2 + 10. The function g is a line with slope 2 that passes through the point (1, 3).
Complete the statements.
f(3) =
g(3) =
At x = 3, f is g.
Choose an answer from each menu.
Hint
For f, square first, then apply the negative sign. For g, start at (1, 3) and move 2 units right, rising 2 for each unit.
Step-by-step solution
f(3) = -(3)^2 + 10 = -9 + 10 = 1.
g: from (1, 3) to x = 3 is 2 units right; with slope 2 the output rises 2 \times 2 = 4. So g(3) = 3 + 4 = 7.
(Or find the equation: 3 = 2(1) + b, so b = 1 and g(x) = 2x + 1; g(3) = 7.)
Compare: 1 < 7, so f is less thang at x = 3.
Answer: f(3) = 1, g(3) = 7; f is less than g at x = 3.
Check:g(1) = 2 + 1 = 3 ✓ matches the given point.
Tip: In -x^2, the square is done first and the minus sign after: -(3)^2 = -9, never +9.
Question 33 of 36
Two small companies track their monthly profit, in thousands of dollars.
Company A:P(x) = -2x^2 + 40x - 50, where x is the number of months since opening.
Company B: had a profit of -20 thousand dollars (a loss) in its opening month (x = 0), and its monthly profit increases by 12 thousand dollars each month.
What is the first whole month x in which Company B's monthly profit is greater than Company A's maximum monthly profit?
Hint
Find Company A's maximum profit using the vertex. Write Company B's profit as a linear function and solve an inequality.
Step-by-step solution
Company A's maximum:a = -2 < 0, so P has a maximum at x = -\frac{40}{2(-2)} = 10.
The first whole month greater than 14.1\overline{6} is x = 15.
Answer: Month 15
Check:B(14) = -20 + 168 = 148, not more than 150 ✗; B(15) = -20 + 180 = 160 > 150 ✓
Tip: With "first whole number greater than," round up, then test the whole numbers on both sides of your answer.
Question 34 of 36
Fill in
A park has two fountains. Each shoots an arc of water from ground level. The height of the water in Fountain A, in feet, at a horizontal distance of x feet from the nozzle is
h(x) = -0.5x^2 + 4x
The arc of Fountain B is a parabola; some of its points are shown in the table.
Distance x (ft)
0
4
6
8
Height (ft)
0
8
9
8
How many feet farther from its nozzle does Fountain B's water land than Fountain A's?
feet
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The water lands where the height is 0 again. Solve h(x) = 0 for Fountain A. For Fountain B, use the symmetry of the table around its highest point.
Step-by-step solution
Fountain A: set -0.5x^2 + 4x = 0. Factor out x: x(-0.5x + 4) = 0.
x = 0 (the nozzle) or -0.5x + 4 = 0 → 0.5x = 4 → x = 8. The water lands 8 ft away.
Fountain B: the heights at x = 4 and x = 8 are both 8, so the axis of symmetry is x = 6 (the peak, 9 ft).
The arc starts at height 0 at x = 0, which is 6 ft left of the axis. By symmetry it lands 6 ft to the right of the axis: x = 6 + 6 = 12.
Difference: 12 - 8 = 4 ft.
Answer: 4 feet
Check: Fountain B fits -0.25x^2 + 3x: at x = 12, -0.25(144) + 36 = -36 + 36 = 0 ✓
Tip: A parabola's two zeros are the same distance from its axis of symmetry. If you know one zero and the vertex, you can find the other zero without an equation.
Question 35 of 36
The function f is defined by f(x) = x^2. The linear function g is shown in the table.
x
0
1
2
g(x)
6
7
8
For which value of x is f(x)less thang(x)?
Hint
Find a rule for g from the table. Then test each choice: compute f(x) and g(x) and compare.
Step-by-step solution
From the table, g starts at 6 and rises 1 per unit: g(x) = x + 6.
Check: Solve x^2 = x + 6: x^2 - x - 6 = (x - 3)(x + 2) = 0, so the graphs meet at x = -2 and x = 3. Between them the line is above the parabola, and 2.5 is between -2 and 3 ✓
Tip: "Less than" is strict: where the functions are equal (x = 3) does not count. Testing each answer choice is often the fastest GED strategy.
Question 36 of 36
Two swimming pools are being drained. The water in Pool A is shown in the table. The water in Pool B, in gallons, after t minutes is V(t) = 10{,}000 - 75t.
Minutes t
15
40
60
Water in Pool A (gal)
9,250
7,250
5,650
Which pool is draining faster, and by how many gallons per minute?
Hint
Look carefully at the time column: the gaps are not equal. Divide each change in water by its own change in time.
Step-by-step solution
Pool A, first interval: \dfrac{7250 - 9250}{40 - 15} = \dfrac{-2000}{25} = -80 gallons per minute.
Pool A, second interval: \dfrac{5650 - 7250}{60 - 40} = \dfrac{-1600}{20} = -80 ✓ (the rate is constant, so Pool A is linear).
Pool B: V(t) = 10{,}000 - 75t drains 75 gallons per minute.
Compare: Pool A drains 80 gal/min, Pool B 75 gal/min. 80 - 75 = 5.
