The equation and the graph represent two linear functions.
Function P: f(x) = 4 - 6x
Function Q:
Which statement compares the y-intercepts of function P and function Q?
- A. The y-intercept of function P is -6 which is less than the y-intercept of function Q.
- B. The y-intercept of function P is 4 which is equal to the y-intercept of function Q.
- C. The y-intercept of function P is -6 which is greater than the y-intercept of function Q.
- D. The y-intercept of function P is 4 which is greater than the y-intercept of function Q.
Correct Answer & Rationale
Correct Answer: D
Function P, represented by the equation \( f(x) = 4 - 6x \), has a y-intercept of 4, which is found by evaluating \( f(0) \). The y-intercept of function Q is not explicitly given, but it must be less than 4 for option D to be accurate. Option A incorrectly states that the y-intercept of P is -6. Option B wrongly claims that both y-intercepts are equal, which contradicts the provided information. Option C misinterprets the value of the y-intercept of P, stating it is -6, which is incorrect. Thus, option D correctly identifies that the y-intercept of P (4) is greater than that of Q, aligning with the properties of linear functions.
Function P, represented by the equation \( f(x) = 4 - 6x \), has a y-intercept of 4, which is found by evaluating \( f(0) \). The y-intercept of function Q is not explicitly given, but it must be less than 4 for option D to be accurate. Option A incorrectly states that the y-intercept of P is -6. Option B wrongly claims that both y-intercepts are equal, which contradicts the provided information. Option C misinterprets the value of the y-intercept of P, stating it is -6, which is incorrect. Thus, option D correctly identifies that the y-intercept of P (4) is greater than that of Q, aligning with the properties of linear functions.
Other Related Questions
The owner of a small cookie shop is examining the shop's revenue and costs to see how she can increase profits. Currently, the shop has expenses of $41.26 and $0.19 per cookie.
The shop's revenue and profit depend on the sales price of the cookies. The daily revenue is given in the graph below, where x is the sales price of the cookies and y is the expected revenue at that price.
The shop owner needs to determine the total daily cost of making x cookies. Which of the following linear equations represents the cost, C, in dollars?
- A. C=4.6x+995
- B. C=0.046x+2
- C. C=0.19x+41.26
- D. C=1.2x+212.26
Correct Answer & Rationale
Correct Answer: C
The equation representing total daily cost must account for both fixed and variable costs. The fixed cost of $41.26 reflects the shop's expenses, while the variable cost is $0.19 per cookie, leading to the term 0.19x for x cookies. Therefore, C = 0.19x + 41.26 accurately captures both components. Option A incorrectly suggests a much higher fixed cost and variable rate, implying unrealistic expenses. Option B has a fixed cost that is too low and a variable cost that is also incorrect. Option D presents exaggerated figures for both fixed and variable costs, misrepresenting the shop's actual expenses.
The equation representing total daily cost must account for both fixed and variable costs. The fixed cost of $41.26 reflects the shop's expenses, while the variable cost is $0.19 per cookie, leading to the term 0.19x for x cookies. Therefore, C = 0.19x + 41.26 accurately captures both components. Option A incorrectly suggests a much higher fixed cost and variable rate, implying unrealistic expenses. Option B has a fixed cost that is too low and a variable cost that is also incorrect. Option D presents exaggerated figures for both fixed and variable costs, misrepresenting the shop's actual expenses.
The mass of an amoeba is approximately 4.0 × 10^(-6) grams. Approximately how many amoebas are present in a sample that weighs 1 gram?
- A. 2.5 × 10^5
- B. 4.0 × 10^7
- C. 4.0 × 10^5
- D. 2.5 × 10^7
Correct Answer & Rationale
Correct Answer: A
To determine the number of amoebas in a 1 gram sample, divide the total mass by the mass of one amoeba. The mass of an amoeba is 4.0 × 10^(-6) grams. Thus, the calculation is: 1 gram / (4.0 × 10^(-6) grams/amoeba) = 2.5 × 10^5 amoebas. Option B (4.0 × 10^7) is incorrect as it suggests a significantly larger quantity, likely resulting from a miscalculation. Option C (4.0 × 10^5) overestimates the number of amoebas by a factor of 2, while option D (2.5 × 10^7) also miscalculates, indicating confusion in the division process.
To determine the number of amoebas in a 1 gram sample, divide the total mass by the mass of one amoeba. The mass of an amoeba is 4.0 × 10^(-6) grams. Thus, the calculation is: 1 gram / (4.0 × 10^(-6) grams/amoeba) = 2.5 × 10^5 amoebas. Option B (4.0 × 10^7) is incorrect as it suggests a significantly larger quantity, likely resulting from a miscalculation. Option C (4.0 × 10^5) overestimates the number of amoebas by a factor of 2, while option D (2.5 × 10^7) also miscalculates, indicating confusion in the division process.
Type your answer in the box. You may use numbers, a decimal point (.), and/or a negative sign (-) in your answer.
A company received a shipment of 8 boxes of metal brackets.
• There are 20 metal brackets in each box.
• The total weight of the shipment is 48 pounds.
What is the weight, in pounds, of each metal bracket?
Correct Answer & Rationale
Correct Answer: 0.3
To find the weight of each metal bracket, first calculate the total number of brackets by multiplying the number of boxes (8) by the number of brackets per box (20), resulting in 160 brackets. Next, divide the total weight of the shipment (48 pounds) by the total number of brackets (160). This calculation yields a weight of 0.3 pounds per bracket. Other options may include numbers that misrepresent the division or assume incorrect values for the total brackets or shipment weight. For example, using a weight of 1 pound per bracket would imply only 48 brackets, which contradicts the initial information provided.
To find the weight of each metal bracket, first calculate the total number of brackets by multiplying the number of boxes (8) by the number of brackets per box (20), resulting in 160 brackets. Next, divide the total weight of the shipment (48 pounds) by the total number of brackets (160). This calculation yields a weight of 0.3 pounds per bracket. Other options may include numbers that misrepresent the division or assume incorrect values for the total brackets or shipment weight. For example, using a weight of 1 pound per bracket would imply only 48 brackets, which contradicts the initial information provided.
Which graph shows 3y - 2x = 6?
- A. M-36A.png
- B. M-36B.png
Correct Answer & Rationale
Correct Answer: B
To determine the graph of the equation \(3y - 2x = 6\), we can rearrange it into slope-intercept form \(y = mx + b\). This gives us \(y = \frac{2}{3}x + 2\), indicating a slope of \(\frac{2}{3}\) and a y-intercept of 2. Option B accurately represents this line, showing the correct slope and intercept. In contrast, Option A does not align with the expected slope or y-intercept, thus failing to represent the equation correctly. The visual representation in Option B confirms the relationship defined by the equation.
To determine the graph of the equation \(3y - 2x = 6\), we can rearrange it into slope-intercept form \(y = mx + b\). This gives us \(y = \frac{2}{3}x + 2\), indicating a slope of \(\frac{2}{3}\) and a y-intercept of 2. Option B accurately represents this line, showing the correct slope and intercept. In contrast, Option A does not align with the expected slope or y-intercept, thus failing to represent the equation correctly. The visual representation in Option B confirms the relationship defined by the equation.