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).

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).

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).

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.

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.

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.

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.

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.

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.

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.