This question asks for the point-slope form of a line with slope -3 passing through point (4,2). The point-slope form is y - y₁ = m(x - x₁), where m is the slope and (x₁,y₁) is the given point. Substituting m = -3 and (x₁,y₁) = (4,2): y - 2 = -3(x - 4). Common errors include sign mistakes when substituting negative coordinates or confusing which coordinate goes where, resulting in forms like y + 2 = -3(x - 4) or y - 4 = -3(x - 2). Remember that in point-slope form, we always subtract the point's coordinates from the variables.

The streaming service charges $12 per month plus a one-time activation fee, and the total after 6 months is $87. Let A be the activation fee; then the total cost equation is: A + 12(6) = 87. Solving: A + 72 = 87, so A = 87 - 72 = 15. The activation fee is $15. A common error is forgetting to multiply the monthly rate by the number of months or including the activation fee multiple times. When solving cost problems with one-time fees, set up an equation that adds the one-time fee once to the recurring charges.

We need to find the equation of the line passing through (2, -1) and (-4, 11) in the form y = mx + b. First, calculate the slope: m = (11 - (-1))/(-4 - 2) = 12/(-6) = -2. Now use point-slope form with (2, -1): y - (-1) = -2(x - 2), which gives y + 1 = -2x + 4, so y = -2x + 3. The most common error is sign mistakes when subtracting negative numbers in the slope calculation. Always double-check by substituting both original points back into your final equation to verify they satisfy it.

This problem describes a water tank scenario where we need to model the total amount of water over time. The tank starts with 15 gallons (initial amount) and fills at 3 gallons per minute (rate of change). In the linear equation y = mx + b, the slope m represents the rate of change (3 gallons/minute) and the y-intercept b represents the initial amount (15 gallons). Therefore, the equation is y = 3x + 15. A common error is confusing the initial amount with the rate of change. Remember that the y-intercept represents the starting value when time x = 0.

The question asks for the value of f(5) for a linear function f where f(2)=7 and f(8)=19. First, calculate the slope m = (19 - 7)/(8 - 2) = 12/6 = 2. Using the point-slope form with (2, 7), f(x) = 2(x - 2) + 7 = 2x - 4 + 7 = 2x + 3. Therefore, f(5) = 2(5) + 3 = 10 + 3 = 13, which is choice C. Alternatively, from x=2 to x=5 is an increase of 3, so with slope 2, the function increases by 3*2=6, giving 7+6=13. A common error is miscalculating the slope, such as dividing 12 by 8-2 incorrectly. As a test-taking strategy, use the constant rate of change in linear functions to interpolate values between given points.

This question asks for f(3) given that f is linear with f(1) = 9 and f(5) = 1. First, find the slope: m = (1 - 9)/(5 - 1) = -8/4 = -2. Since f is linear, it decreases by 2 for each unit increase in x. From x = 1 to x = 3 is an increase of 2 units, so f(3) = f(1) + 2(-2) = 9 - 4 = 5. A common error is computing the slope correctly but then applying it from the wrong starting point, such as calculating from x = 5 instead of x = 1. When finding intermediate values of linear functions, work from the nearest known point to minimize calculation steps.

This question asks for the interpretation of the y-intercept in the context of a taxi fare model C = 2.00m + 4.50. In this equation, C is the total cost, m is the number of miles, 2.00 is the cost per mile (slope), and 4.50 is the y-intercept. The y-intercept represents the value of C when m = 0, which is the base fare charged before any miles are driven. This is the fixed cost that every passenger pays regardless of distance. Students often confuse the y-intercept with other quantities like the rate of change or incorrectly interpret it as a distance value. In real-world linear models, the y-intercept always represents the initial or fixed value when the independent variable equals zero.

This problem asks about interpreting the y-intercept in a linear cost function context. The equation for this situation would be $y = 20x + 45$, where x represents months and y represents total cost. In slope-intercept form $y = mx + b$, the y-intercept b represents the value of y when $x = 0$. Here, when $x = 0$ months, the total cost y would be $45, which corresponds to the one-time sign-up fee paid before any monthly charges. The y-intercept represents the initial value or starting point of the linear relationship. In cost problems, the y-intercept typically represents fixed or upfront costs rather than ongoing rates.

The question asks for the value of f(-2) where f(x) = -4x + 9. Substitute x = -2 into the function: f(-2) = -4(-2) + 9 = 8 + 9 = 17. This linear function has a slope of -4 and y-intercept 9, so evaluating at a negative x yields a positive adjustment. Verify by considering the line's behavior: as x decreases, y increases due to the negative slope. A common error is forgetting the negative sign in -4(-2), leading to -4(-2) = -8 instead of 8, resulting in -8 + 9 = 1. Another mistake might be misapplying order of operations or substituting incorrectly. As a test-taking strategy, always double-check sign changes when evaluating functions with negative inputs.

This problem asks for the point-slope form of a line given a point and slope. Point-slope form is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the known point. Given point (3, -7) and slope 4, we substitute: y - (-7) = 4(x - 3), which simplifies to y + 7 = 4(x - 3). This matches choice A exactly. The key is to substitute the coordinates correctly: subtract the y-coordinate from y and subtract the x-coordinate from x, being careful with signs. A common error is confusing positive and negative signs when the coordinates are negative.