Skip to main content
Sign in Create a free account
Varsity Tutors
Sign in

Interpreting Rate & Initial Value - Math 1

  1. If the company increases the per-mile rate by $0.15 but keeps the base fee unchanged, what will be the new initial value when no miles are driven?
    Answer: $45.00 The initial value (y-intercept) represents the cost when m = 0, which is the base fee. The original equation is C = 0.35m + 45, where 45 is the base fee. If only the per-mile rate increases by $0.15, the new equation becomes C = 0.50m + 45. The base fee (initial value) remains $45.00. Choice A might result from incorrectly subtracting something from the base fee. Choice C might come from incorrectly adding the rate increase to the base fee. Choice D represents the new per-mile rate, not the initial value.
  2. The function f(x)=−3x+bf(x) = -3x + b passes through the point (4,−5)(4, -5). If the function is shifted up by 7 units to create g(x)g(x), what is the rate of change of g(x)g(x)?
    Answer: -3 The rate of change of a linear function is its slope, which remains unchanged when the function is shifted vertically. Since f(x) = -3x + b has a slope of -3, and g(x) is f(x) shifted up by 7 units, g(x) = -3x + b + 7. The rate of change of g(x) is still -3. Choice B (4) might be chosen by confusing the x-coordinate of the given point with the rate. Choice C (7) represents the vertical shift amount. Choice D (10) might result from incorrectly adding the shift to something else.
  3. A water tank is being filled at a constant rate. After 3 minutes, the tank contains 45 gallons. After 8 minutes, the tank contains 70 gallons. What was the initial amount of water in the tank?
    Answer: 30 gallons First find the rate: (70-45)/(8-3) = 25/5 = 5 gallons per minute. Then use point-slope form with (3,45): amount = 5t + b. Substituting: 45 = 5(3) + b, so 45 = 15 + b, therefore b = 30 gallons initially. Choice A results from subtracting the rate from the 3-minute amount. Choice B comes from incorrectly using the 8-minute data point. Choice C results from averaging the two given amounts.
  4. A function is defined by the equation f(x)=−2x+kf(x) = -2x + k, where k is a constant. If f(4)=3f(4) = 3, what is the initial value of the function g(x)=f(x−2)g(x) = f(x-2)?
    Answer: 15 First find k: f(4) = 3 means -2(4) + k = 3, so -8 + k = 3, thus k = 11. So f(x) = -2x + 11. Now g(x) = f(x-2) = -2(x-2) + 11 = -2x + 4 + 11 = -2x + 15. The initial value of g(x) is 15. Choice A (7) might result from errors in the horizontal shift. Choice B (11) is the value of k, the initial value of f(x). Choice D (19) might come from incorrectly adding values.
  5. The equation y=−2.5x+40y = -2.5x + 40 models the temperature inside a refrigerator during a power outage, where xx is time in hours and yy is temperature in degrees Fahrenheit. In this context, what does the coefficient -2.5 represent?
    Answer: The temperature decreases by 2.5 degrees every hour during the outage In the linear equation y = mx + b, the coefficient m represents the rate of change. Since m = -2.5, the temperature decreases by 2.5 degrees per hour. The negative sign indicates decrease. Choice B ignores the negative sign. Choice C confuses the rate with the initial value (which is 40). Choice D misinterprets the coefficient as a final temperature value.
  6. If someone observes that after 2 hours the candle is 6.5 inches tall, how does this compare to the model's prediction, and what would this suggest about the actual initial height?
    Answer: The model predicts 6.5 inches; the observation matches perfectly with an initial height of 8 inches When you see a linear function modeling a real-world situation, you need to evaluate the model by substituting given values and comparing predictions to observations. Let's check what the model predicts after 2 hours. Substituting t=2t = 2 into h(t)=8−0.75th(t) = 8 - 0.75t: h(2)=8−0.75(2)=8−1.5=6.5h(2) = 8 - 0.75(2) = 8 - 1.5 = 6.5 inches The model predicts exactly 6.5 inches, which matches the observation perfectly. Since the observation aligns with the model, this confirms that the initial height assumption of 8 inches (when t=0t = 0) is correct. Looking at the wrong answers: Choice A incorrectly states that the observation is lower than the prediction, when they're actually equal. Choice B incorrectly claims the observation is higher than the prediction. Both A and B misread the relationship between prediction and observation. Choice D makes a calculation error—it states the model predicts 5.5 inches instead of the correct 6.5 inches, possibly from miscalculating 8−0.75(2)8 - 0.75(2). The key insight is that when real-world data matches a mathematical model's predictions, it validates the model's parameters, including the initial conditions. Study tip: When evaluating linear models, always substitute the given values carefully and compare numerical results exactly. If observations match predictions, the model parameters are likely correct. If they don't match, you'll need to adjust parameters like the initial value or rate of change.
  7. A linear function has the equation y=mx+3y = mx + 3. If the function passes through the point (6,−9)(6, -9), what is the initial value of the function y=2mx+7y = 2mx + 7?
    Answer: 7 This question tests your understanding of linear functions and how to find the initial value (y-intercept) when given different slope coefficients. First, you need to find the value of mm from the given information. Since y=mx+3y = mx + 3 passes through (6,−9)(6, -9), substitute these coordinates: −9=m(6)+3-9 = m(6) + 3. Solving for mm: −9=6m+3-9 = 6m + 3, so −12=6m-12 = 6m, which gives us m=−2m = -2. Now you can find the initial value of y=2mx+7y = 2mx + 7. The initial value is the y-intercept, which occurs when x=0x = 0. Substituting m=−2m = -2 into the second function: y=2(−2)x+7=−4x+7y = 2(-2)x + 7 = -4x + 7. When x=0x = 0, y=7y = 7. Looking at the wrong answers: Choice A gives -2, which is the value of mm, not the initial value of the second function. This confuses the slope with the y-intercept. Choice B gives 3, which is the initial value of the first function y=mx+3y = mx + 3, not the second function. Choice C gives -9, which is the y-coordinate of the given point, but this isn't relevant to finding the initial value of the new function. Remember that the initial value of any linear function y=mx+by = mx + b is always the constant term bb, regardless of what mm equals. Don't let the coefficient of xx distract you from identifying the y-intercept.
  8. What does the coefficient 72 represent in the context of this cooling model?
    Answer: The room temperature that the object approaches as time increases As t approaches infinity, e−0.2te^{-0.2t} approaches 0, so T approaches 72. This represents the ambient temperature the object approaches. At t = 0, T = 72 + 150(1) = 222°F, which is the initial temperature. Choice A (initial temperature) would be 222°F, not 72. Choice C (cooling rate) is related to the -0.2 coefficient in the exponent. Choice D (temperature difference) would be 150, the coefficient of the exponential term.
  9. Based on this model, which interpretation is correct about the phone's battery usage?
    Answer: The phone started with 85% battery and loses 12% per hour while streaming In P = 85 - 12t, the initial value is 85% (when t=0) and the rate of change is -12% per hour. Choice A reverses the initial value and rate. Choice C incorrectly uses the initial percentage as time duration. Choice D misinterprets the negative rate as positive (gaining battery).
  10. What is the new initial value after the equipment upgrades?
    Answer: $3000 The original function is C = 150n + 2400, where 150 is the variable cost per item and 2400 is the fixed cost (initial value). After upgrades: new variable cost = 150 - 25 = 125 per item, and new fixed cost = 2400 + 600 = 3000. The new function is C = 125n + 3000. The initial value is $3000. Choice A ($2400) is the original initial value. Choice B ($2775) might result from calculation errors. Choice D ($3025) might come from incorrectly including the variable cost change in the fixed cost.
  11. After 3 hours, the growth rate doubles while the current population becomes the new baseline. What is the initial value of the new growth function?
    Answer: 650 After 3 hours, P(3) = 200 + 150(3) = 200 + 450 = 650. This becomes the new baseline (initial value) for the new function. The original growth rate was 150 per hour, so the doubled rate is 300 per hour. The new function starts at t = 3 with population 650, so if we reset time to t = 0 at this point, the new function is P_new(t) = 650 + 300t, with initial value 650. Choice A (200) is the original initial value. Choice B (300) is the new growth rate. Choice D (950) might result from adding another time period incorrectly.
  12. A savings account earns simple interest over time. The account balance can be represented by B(t)=500+15tB(t) = 500 + 15t, where B(t)B(t) is the balance in dollars and tt is time in months. After 18 months, the account holder makes a one-time deposit of $200. What is the new rate of change and initial value for the account balance after this deposit?
    Answer: Rate of change: $15 per month; Initial value: $700 The one-time deposit only shifts the function vertically by $200, changing it to B(t) = 700 + 15t. The rate of change remains $15 per month, but the initial value becomes $500 + $200 = $700. Choice A uses the original function. Choice C incorrectly adds the deposit to the rate. Choice D incorrectly calculates both the rate and initial value.

Start this quiz