- Function u is given by the equation u(x)=−x+9. Function v is shown as a line passing through the points (0,6) and (3,0). At x=2, which function has the greater output value?
Answer: Function u, because u(2)=7 and v(2)=2.
This problem tests comparing linear functions from different representations (equation and graph) by evaluating both at x = 2. From equation u(x) = -x + 9, we calculate u(2) = -2 + 9 = 7. For function v shown as a line through (0, 6) and (3, 0), we first find its equation: slope = (0-6)/(3-0) = -6/3 = -2, y-intercept = 6, so v(x) = -2x + 6. Then v(2) = -2(2) + 6 = -4 + 6 = 2. Comparing outputs at x = 2: u(2) = 7 and v(2) = 2, so 7 > 2, meaning function u has the greater output value. Choice A incorrectly reverses the comparison, Choice C incorrectly states v's y-intercept as 9, and Choice D incorrectly claims equal values. The strategy is to evaluate both functions at the given x-value and compare directly. - Function A is shown in the table. Function B is described as: "It starts at 5 when x=0 and increases by 3 for every 1 increase in x."
Which function grows faster?
Answer: Function B, because its rate of change is 3 and Function A's is 2
This question tests comparing growth rates (slopes) between a table and a verbal description. For Function A from the table, we calculate slope by finding Δy/Δx between consecutive points; if the pattern shows slope 2, that's the rate of change. Function B is described verbally: "starts at 5 when x=0" gives y-intercept=5, and "increases by 3 for every 1 increase in x" means slope=3. Comparing growth rates: Function A has slope 2 and Function B has slope 3, so since 3>2, Function B grows faster. The error in option A focuses on which starts higher (y-intercept) rather than growth rate, while option B confuses the rate of change (3) with initial value. Strategy: (1) recognize "grows faster" means compare slopes, (2) calculate slope from table, (3) extract slope from verbal description ("increases by ___ per ___"), (4) compare numerical slopes, (5) distinguish between starting value and rate of increase.
- Function A is given by f(x)=−2x+10. Function B is shown in the table.
Which statement is true about their rates of change?
Answer: Function B has a greater rate of change because 3>−2
This question tests comparing rates of change (slopes) between an equation and a table representation. From Function A's equation f(x)=-2x+10, we identify the slope as -2 (coefficient of x). For Function B's table, we calculate slope using consecutive points: if the table shows values like (0,0), (1,3), (2,6), then slope = (3-0)/(1-0) = 3. Comparing the slopes: Function A has slope -2 and Function B has slope 3, and since 3>-2 (positive is greater than negative), Function B has the greater rate of change. The error in option A compares the y-intercepts (10 vs some value) instead of slopes, while option C incorrectly claims -2>3. Strategy: (1) extract slope from equation (coefficient of x, including sign), (2) calculate slope from table using Δy/Δx, (3) compare signed numbers correctly (positive > negative), (4) remember that "greater rate of change" means larger slope value, not steeper decline. - Function A is given by the equation f(x)=3x+2. Function B is shown in the table.
Which function has the greater rate of change (slope)?
Answer: Function B, because its rate of change is 5 and 5>3
This question tests comparing linear functions from different representations (equation and table) by extracting and comparing rate of change (slope). From Function A's equation f(x)=3x+2, we directly identify the slope as 3 (coefficient of x) and y-intercept as 2. For Function B's table, we calculate slope using Δy/Δx between any two points: using (0,1) and (1,6), slope = (6-1)/(1-0) = 5/1 = 5, and the y-intercept is 1 (value when x=0). Comparing the slopes: Function B has slope 5 and Function A has slope 3, so since 5>3, Function B has the greater rate of change. The error in option C confuses the y-intercept value (2) with the slope, while option A incorrectly focuses on the y-intercept instead of rate of change. Strategy: (1) identify that rate of change means slope, (2) extract slope from equation (coefficient of x), (3) calculate slope from table (Δy/Δx), (4) compare numerical values, (5) select function with larger slope. - Function A is given by f(x)=2x+9. Function B is described as: "It starts at 12 when x=0 and decreases by 1 for each increase of 1 in x." At x=4, which function has the greater value?
Answer: Function A, because f(4)=17 and Function B is 8 at x=4
This question tests comparing linear functions from different representations (equation, table, graph, verbal) by extracting and comparing rate of change (slope) and initial value (y-intercept). Extract properties: from y=mx+b equation (m=slope, b=intercept directly), from table (slope=Δy/Δx between rows: (6-1)/(1-0)=5, intercept=y when x=0), from graph (slope=rise/run counting grid squares, intercept where crosses y-axis), from verbal ("starts at 5"=intercept, "increases by 3 per"=slope). Compare: larger slope grows faster (steeper), larger intercept starts higher. For example, to compare at x=4, equation f(x)=2x+9 gives 2*4+9=17, verbal starts at 12 decreases by 1 per x so 12-4=8, comparison 17>8 so equation is greater. In this question, Function A at x=4 is 17, Function B at x=4 is 8, so Function A has the greater value because 17>8. A common error is miscalculating the verbal function like subtracting wrong or confusing decrease with increase. Strategy: (1) identify representation type for each function, (2) extract slope (equation: coefficient of x, table: Δy/Δx, graph: rise/run, verbal: rate stated), (3) extract y-intercept (equation: constant term, table: y at x=0, graph: y-axis crossing, verbal: initial value), (4) compare (which m larger? which b larger?), (5) interpret (steeper slope means faster growth, higher intercept means higher start). Common errors: confusing slope and intercept (using b value as rate), inverting slope from table (Δx/Δy), misreading graph (counting wrong or reading wrong point), misinterpreting verbal (rate vs initial value confused). - Function A is shown in the table. Function B is given by the equation g(x)=4x−1.
Which function has the greater initial value (the y-intercept)?
Answer: Function A, because its initial value is 7 and 7>−1
This question tests comparing initial values (y-intercepts) of linear functions from different representations (table and equation). From Function B's equation g(x)=4x-1, we identify the slope as 4 and y-intercept as -1 (the constant term). For Function A's table, we need to find the y-intercept by looking at the value when x=0, which appears to be 7 based on the pattern (though the table isn't shown, this is implied by the correct answer). Comparing initial values: Function A has y-intercept 7 and Function B has y-intercept -1, so since 7>-1, Function A has the greater initial value. The errors in options B and D confuse slope (4) with initial value, while option C incorrectly identifies 7 as the slope rather than the y-intercept. Strategy: (1) recognize initial value means y-intercept, (2) extract y-intercept from equation (constant term), (3) find y-intercept from table (y-value when x=0), (4) compare values including negative numbers correctly, (5) avoid confusing slope and intercept. - Function A is described as: "Starts at 5 and increases by 2 for every 1 increase in x."
Function B is given by the equation h(x)=3x+1.
Which function grows faster as x increases?
Answer: Function B, because its slope is 3 and Function A's rate is 2
Tests comparing linear functions from different representations (equation, table, graph, verbal) by extracting and comparing rate of change (slope) and initial value (y-intercept). Extract properties: from y=mx+b equation (m=slope, b=intercept directly), from table (slope=Δy/Δx between rows: (6-1)/(1-0)=5, intercept=y when x=0), from graph (slope=rise/run counting grid squares, intercept where crosses y-axis), from verbal ("starts at 5"=intercept, "increases by 3 per"=slope). Compare: larger slope grows faster (steeper), larger intercept starts higher. For example, Function A described verbally starts at 5 with rate 2 (slope 2, intercept 5), while the equation h(x)=3x+1 has slope 3 and intercept 1, so Function B grows faster since 3>2. In this question, Function B grows faster as x increases because its slope of 3 is greater than Function A's rate of 2. A common error is misinterpreting the verbal description, such as thinking Function A's starting value of 5 is its slope, or reversing the comparison by claiming 2>3. Strategy: (1) identify representation type for each function, (2) extract slope (equation: coefficient of x, table: Δy/Δx, graph: rise/run, verbal: rate stated), (3) extract y-intercept (equation: constant term, table: y at x=0, graph: y-axis crossing, verbal: initial value), (4) compare (which m larger? which b larger?), (5) interpret (steeper slope means faster growth, higher intercept means higher start). Common errors: confusing slope and intercept (using b value as rate), inverting slope from table (Δx/Δy), misreading graph (counting wrong or reading wrong point), misinterpreting verbal (rate vs initial value confused).
- Function A is given by f(x)=6x−4. Function B is described as: "It starts at 2 when x=0 and decreases by 1 for every 1 increase in x."
Which statement is true?
Answer: Function A has the greater rate of change because 6>−1
This question tests comparing properties of functions given as an equation and a verbal description. Function A from f(x)=6x-4 has slope 6 and y-intercept -4. Function B is described as "starts at 2 when x=0" (y-intercept=2) and "decreases by 1 for every 1 increase in x" (slope=-1, negative because it decreases). Comparing rates of change: Function A has slope 6 and Function B has slope -1, so 6>-1, meaning Function A has the greater rate of change. Option A incorrectly compares initial values instead of rates, option C makes a false comparison (-4>2 is false), and option D is incorrect as the slopes differ. Strategy: (1) extract slope and intercept from equation, (2) interpret verbal description carefully ("decreases by 1" means slope -1), (3) compare slopes as signed numbers, (4) remember positive slope > negative slope regardless of magnitude. - Function r is given by r(x)=5x−4. Function s is described as: "It starts at 1 when x=0 and increases by 3 for every 1 increase in x."
Which statement correctly compares their rates of change?
Answer: Function r has a greater rate of change than function s.
This question tests comparing rates of change between equation and verbal representations. Function r has equation r(x) = 5x - 4, so its rate of change (slope) is 5. Function s is described as increasing by 3 for every 1 increase in x, so its rate of change is 3. Comparing: 5 > 3, so function r has the greater rate of change, making answer A correct. The initial values (r starts at -4, s starts at 1) are not relevant for this comparison. A common error would be comparing the initial values instead of the rates, or misinterpreting "increases by 3" as meaning the initial value rather than the slope. - The graph represents function m. Function n is given by n(x)=32x+1. At what value of x do these functions have the same output value?
Answer: x=6, and at this point both functions have an output value of 5
From the graph, function m appears to pass through (0, -1) and (3, 2), giving it slope (2-(-1))/(3-0) = 1 and equation m(x) = x - 1. Setting m(x) = n(x): x - 1 = (2/3)x + 1. Solving: x - (2/3)x = 2, so (1/3)x = 2, thus x = 6. At x = 6: n(6) = (2/3)(6) + 1 = 4 + 1 = 5, and m(6) = 6 - 1 = 5. So they intersect at (6, 5). - Function u is shown in the graph. Function v is represented by the table shown. Over the interval from x=1 to x=5, which function has the greater average rate of change, and by how much?
Answer: Function u has the greater average rate of change by exactly 21 unit per unit
From the graph, u(1) = 2 and u(5) = 6. Average rate of change for u = (6-2)/(5-1) = 4/4 = 1. From the table, v(1) = 3 and v(5) = 5. Average rate of change for v = (5-3)/(5-1) = 2/4 = 1/2. Function u has the greater rate by 1 - 1/2 = 1/2 unit per unit. - Function a is shown on the coordinate plane as a line passing through the points (0,4) and (2,8). Function b is given by the equation b(x)=5x+1. Which statement is true?
Answer: Function b has the greater rate of change, and function a has the greater y-intercept.
This problem tests comparing linear functions from different representations (graph and equation) by extracting both rate of change and y-intercept. From the graph, function a passes through (0, 4) and (2, 8), so slope = rise/run = (8-4)/(2-0) = 4/2 = 2, and y-intercept = 4 (the y-value when x = 0). From equation b(x) = 5x + 1, we extract slope = 5 and y-intercept = 1. Comparing rates of change: 5 > 2, so b has the greater rate of change. Comparing y-intercepts: 4 > 1, so a has the greater y-intercept. Therefore, the correct statement is that function b has the greater rate of change, and function a has the greater y-intercept. Common errors include miscalculating slope from the graph or confusing which function has which property. - Function a is described as: "It starts at 6 when x=0 and increases by 3 for every 1 increase in x."
Function b is given by b(x)=2x+9.
Which statement correctly compares their initial values?
Answer: Function b has a greater initial value than function a.
This question tests comparing initial values between verbal and equation representations. Function a is described as starting at 6 when x = 0, so its initial value is 6. Function b is given by b(x) = 2x + 9, where the initial value (y-intercept) is 9. Comparing these: 9 > 6, so function b has the greater initial value, making answer B correct. The verbal description "increases by 3 for every 1 increase in x" tells us the slope is 3, but this is not relevant for comparing initial values. A common error would be confusing the rate of change (3) with the initial value (6) in the verbal description, or using the slope coefficient (2) instead of the y-intercept (9) from the equation. - Function A is given by f(x)=2.5x+3. Function B is shown in the table.
At x=4, which function has the greater output value?
Answer: Function A, because f(4)=13 and Function B gives 12
This question tests evaluating and comparing function outputs at a specific x-value using different representations. For Function A with f(x)=2.5x+3, we calculate f(4)=2.5(4)+3=10+3=13. For Function B from the table, we need to find the output when x=4; if the table shows a pattern with slope 3 and y-intercept 0 (as suggested by option B), then at x=4, y=3(4)+0=12. Comparing outputs at x=4: Function A gives 13 and Function B gives 12, so Function A has the greater output value since 13>12. The errors in options B and C focus on properties (initial value and rate of change) rather than the specific output at x=4. Strategy: (1) substitute x=4 into the equation, (2) find or calculate the y-value at x=4 from the table, (3) compare the numerical outputs, (4) avoid being distracted by slope or intercept comparisons when asked for a specific value. - Function A is shown in the table. Function B is given by g(x)=2x+1.
At x=3, which function has the greater value?
Answer: Function B, because g(3)=7 and Function A gives 6
This question tests evaluating and comparing function values at a specific x-value from different representations. For Function B with g(x)=2x+1, we calculate g(3)=2(3)+1=6+1=7. For Function A from the table, we need to find the output when x=3; based on the correct answer, the table must show that when x=3, y=6. Comparing at x=3: Function A gives 6 and Function B gives 7, so Function B has the greater value since 7>6. Option A incorrectly claims Function A has greater slope without calculating the specific values at x=3, while option C focuses on initial value rather than the value at x=3. Strategy: (1) substitute x=3 into the equation, (2) find the y-value at x=3 from the table, (3) compare the numerical outputs at that specific x-value, (4) avoid comparing slopes or intercepts when asked for a specific point. - Function A is given by f(x)=x+9. Function B is described as: "It starts at 3 when x=0 and increases by 4 for every 1 increase in x."
Which function will have the greater output for large values of x (for example, x=100)?
Answer: Function B, because its rate of change is greater (4 is greater than 1)
This question tests understanding how different rates of change affect function outputs for large x-values. Function A from f(x)=x+9 has slope 1 and y-intercept 9. Function B is described as starting at 3 (y-intercept=3) and increasing by 4 per unit (slope=4). For large x-values, the function with greater slope will eventually have greater outputs regardless of starting values. At x=100: Function A gives 100+9=109, while Function B gives 4(100)+3=403. Since 403>109, Function B has greater output for large x, confirming that its greater slope (4>1) dominates. Option A incorrectly focuses only on starting values, while option D confuses the y-intercept (9) with the slope. Strategy: (1) identify slopes from both representations, (2) recognize that for large x, slope dominates over y-intercept, (3) verify with a calculation if needed, (4) understand that higher rate of change means steeper growth long-term.
- Function A is given by the equation p(x)=1.5x+6.
Function B is described as: "Starts at 4 and increases by 2 each time x increases by 1."
Which statement correctly compares their initial values and rates of change?
Answer: Function A starts higher, but Function B grows faster
Tests comparing linear functions from different representations (equation, table, graph, verbal) by extracting and comparing rate of change (slope) and initial value (y-intercept). Extract properties: from y=mx+b equation (m=slope, b=intercept directly), from table (slope=Δy/Δx between rows: (6-1)/(1-0)=5, intercept=y when x=0), from graph (slope=rise/run counting grid squares, intercept where crosses y-axis), from verbal ("starts at 5"=intercept, "increases by 3 per"=slope). Compare: larger slope grows faster (steeper), larger intercept starts higher. For example, the equation p(x)=1.5x+6 has slope 1.5 and intercept 6, while Function B described verbally starts at 4 with rate 2 (slope 2, intercept 4), so Function A starts higher (6>4) but Function B grows faster (2>1.5). In this question, the correct statement is that Function A starts higher but Function B grows faster, based on comparing intercepts (6>4) and slopes (1.5<2). A common error is confusing which function has the greater slope or intercept, such as claiming Function A grows faster despite its smaller slope. Strategy: (1) identify representation type for each function, (2) extract slope (equation: coefficient of x, table: Δy/Δx, graph: rise/run, verbal: rate stated), (3) extract y-intercept (equation: constant term, table: y at x=0, graph: y-axis crossing, verbal: initial value), (4) compare (which m larger? which b larger?), (5) interpret (steeper slope means faster growth, higher intercept means higher start). Common errors: confusing slope and intercept (using b value as rate), inverting slope from table (Δx/Δy), misreading graph (counting wrong or reading wrong point), misinterpreting verbal (rate vs initial value confused).
