Interpret Slope And Intercepts - Praxis Core Math

Based on the information in the passage, what is the meaning of the slope of the line that models the freezer's temperature, T, as a function of time, h, in hours after 1:00 PM?
Answer: The temperature increases by 5°C per hour. The slope is the rate of change, calculated as (change in temperature) / (change in time). The temperature changed from -4°C to 6°C, a change of $6 - (-4) = 10$ °C. The time changed from 1:00 PM to 3:00 PM, a change of 2 hours. The slope is $\frac{10°C}{2 \text{ hours}} = 5$ °C per hour.
A sales associate earns a commission based on their monthly sales. The associate's total earnings, E, are modeled by $E = 0.08S + 1500$ , where S is the total monthly sales in dollars. What does the slope of the equation represent?
Answer: The associate's commission rate on sales. The slope of the line is 0.08. Since E represents earnings and S represents sales, the slope is the change in earnings per one-dollar change in sales. This is the commission rate. For every dollar of sales, the associate earns $0.08, which is an 8% commission.
The value of a collectible item, V, after t years is modeled by $V = 75t + 500$ . A second collectible's value is modeled by $V = 100t + 500$ . What is the significance of the fact that both models have the same V-intercept?
Answer: Both collectibles had the same initial value when they were acquired. The V-intercept (or y-intercept) represents the value at t=0, which is the initial value. Since both equations have a V-intercept of 500, it means both items started with a value of $500. Their slopes (rates of increase) are different (75 and 100).
The line shown below represents the total cost $C$ (in dollars) of renting a kayak for $h$ hours. Based on the graph, which equation correctly models the cost?
Answer: $C = 10h + 15$ The y-intercept (at $h=0$ ) is 15 (a flat fee of $15). The line passes through $(0,15)$ and $(4,55)$ , giving slope $\frac{55-15}{4-0}=10$ . So $C=10h+15$ . (A) ignores the y-intercept. (C) swaps slope and intercept. (D) uses rise-over-wrong-run, computing $\frac{55-15}{0-(-4)}$ incorrectly or halving the slope.
The equation of the line graphed below can be written in the form $Ax + By = 12$ . Use the graph to determine the value of $A + B$ .
Answer: $7$ The line has x-intercept $(3,0)$ and y-intercept $(0,4)$ . Using intercept form: $\frac{x}{3}+\frac{y}{4}=1$ , multiply by 12: $4x+3y=12$ . So $A=4, B=3$ , and $A+B=7$ . (A) comes from adding 3+... misidentifying one intercept. (C) comes from using slope $-\tfrac{4}{3}$ as $A/B$ with wrong signs. (D) comes from $4-3=1$ .
A baker's daily profit, P, from selling c cupcakes is given by $P = 1.75c - 80$ . What does the value 80 represent in this equation?
Answer: The daily fixed costs the baker must pay, regardless of sales. The equation is in the form $P = mc + b$ , where b is the y-intercept. Here, the y-intercept is -80. This is the profit when $c=0$ cupcakes are sold. A profit of -80 means a loss of $80. This loss represents the fixed costs (rent, electricity, etc.) that must be paid even if no cupcakes are sold.
A linear equation has a negative slope and a positive y-intercept. If this equation models a company's profit (y) as a function of the number of employees hired (x), what is a possible interpretation of this model?
Answer: The company has some initial income, but each new employee hired results in a net loss of profit. A positive y-intercept means that with zero employees (x=0), the company has a positive profit (y>0), perhaps from investments or automated sales. A negative slope means that as x increases (more employees are hired), y decreases (profit goes down). This implies the cost of hiring an employee is greater than the revenue they generate.
The graph of a linear function $g$ is shown below. Refer to the figure below. Which of the following is the equation of $g$ ?
Answer: $g(x) = -\tfrac{2}{3}x + 2$ The line passes through $(0,2)$ and $(3,0)$ . Slope $=\frac{0-2}{3-0}=-\tfrac{2}{3}$ ; y-intercept is 2. Thus $g(x)=-\tfrac{2}{3}x+2$ . (B) inverts rise and run. (C) ignores the negative direction. (D) misreads the y-intercept as 3 (the x-intercept).
A phone's battery percentage, P, after h hours of use is modeled by $P = -12h + 100$ . If the phone is not recharged, what is the meaning of the h-intercept?
Answer: The number of hours of use after which the battery will be fully depleted. The h-intercept is the value of h when P=0. Setting P=0 gives $0 = -12h + 100$ , so $12h = 100$ , and $h = 100/12 \approx 8.33$ . This value of h represents the time in hours when the battery percentage P is 0, meaning the battery is fully depleted.
For any single day, the cost, C, to park a car in a downtown garage is a flat fee of $20. Which statement best describes the slope of the line representing this cost function, where h is the number of hours parked?
Answer: The slope is 0, because the cost does not change with the number of hours. The cost is constant at $20, regardless of the hours parked. The equation for this is $C = 20$ . This is the equation of a horizontal line. The slope of any horizontal line is 0, indicating there is no rate of change.
The equation $C = 1.25g + 2$ represents the cost, C, of a frozen yogurt based on its weight in ounces, g. Which statement best interprets the y-intercept of this linear model?
Answer: The cost of the cup or cone, regardless of the amount of yogurt. The y-intercept is the value of C when g=0, which is 2. In this context, a weight of zero ounces (g=0) still results in a cost of $2. This represents a fixed charge that is not dependent on weight, such as the cost of the physical cup or cone.
A line passes through the points (2, 100) and (5, 40). If this line models the remaining distance, y, in miles, of a car trip after x hours, what is the correct interpretation of the slope?
Answer: The car's remaining distance decreases by 20 miles each hour. The slope is the change in y divided by the change in x: $m = \frac{40 - 100}{5 - 2} = \frac{-60}{3} = -20$ . Since y represents the remaining distance and x represents hours, the slope of -20 means that the remaining distance decreases by 20 miles for each hour that passes. This is different from speed, which would be positive.
The linear model $y = -4x + 180$ describes the remaining pages, y, a student has to read in a book after x days. What is the interpretation of the x-intercept?
Answer: The total number of days the student will take to finish the book. The x-intercept is the value of x when y=0. Setting y=0 gives $0 = -4x + 180$ , which means $4x = 180$ , so $x = 45$ . Since y represents the pages remaining, y=0 means the book is finished. Therefore, x=45 is the number of days it takes to finish the book.
The amount of water, W, in gallons, in a swimming pool being filled is given by $W = 1200 + 40t$ , where t is the time in minutes. If the pump is replaced with a more powerful one that fills the pool at 60 gallons per minute, how would the graph of the new equation relate to the original graph?
Answer: The new line would be steeper than the original line. The rate at which the pool is filled is the slope of the line. The original slope is 40. The new slope is 60. Since 60 > 40, the new line has a greater slope, which means its graph will be steeper. The y-intercept (the initial amount of water) remains unchanged at 1200.
The amount of money, A, in a savings account after w weeks is modeled by the equation $A = 35w + 250$ . If the initial deposit had been $100 less, but the weekly deposit amount remained the same, what part of the linear model would change?
Answer: The y-intercept would decrease. The initial deposit is the amount of money at week 0, which corresponds to the y-intercept of the graph. In this equation, the y-intercept is 250. The weekly deposit is the rate of change, which is the slope (35). If the initial deposit was $100 less, the y-intercept would decrease to 150, while the slope would remain unchanged.
The height H (in meters) of a hot air balloon is given by $H = 2t + 300$ , where t is in minutes. If the balloon had been launched from a platform 50 meters lower but rose at the same rate, what would be the new equation for the balloon's height?
Answer: $H = 2t + 250$ The initial launch height is the y-intercept of the equation. The rate of rise is the slope. The problem states the balloon rose at the same rate, so the slope (2) remains unchanged. If it launched from 50 meters lower, the initial height (y-intercept) would be $300 - 50 = 250$ . Therefore, the new equation is $H = 2t + 250$ .
The coordinate plane below shows two lines, $\ell_1$ and $\ell_2$ , that are perpendicular. Refer to the figure. If $\ell_1$ has the equation $y = \tfrac{3}{4}x - 2$ and $\ell_2$ passes through the point $(6, 1)$ shown on the graph, what is the y-intercept of $\ell_2$ ?
Answer: $9$ Perpendicular slope: $-\tfrac{4}{3}$ . Using point $(6,1)$ : $y-1=-\tfrac{4}{3}(x-6)\Rightarrow y=-\tfrac{4}{3}x+8+1=-\tfrac{4}{3}x+9$ . Y-intercept is 9. (B) uses slope $\tfrac{4}{3}$ (forgetting the negative). (C) uses slope $-\tfrac{3}{4}$ (negating without reciprocating). (D) uses slope $\tfrac{3}{4}$ (the parallel slope).
The graph below shows the distance $d$ (in miles) a cyclist is from home over time $t$ (in hours). Refer to the graph. During which segment is the cyclist traveling away from home at the fastest rate?
Answer: Segment A (from $t=0$ to $t=1$ ) Slopes: A: $(10-0)/(1-0)=10$ mph away. B: $(10-10)/(2-1)=0$ (stopped). C: $(22-10)/(3.5-2)=8$ mph away. D: $(0-22)/(5-3.5)\approx -14.7$ mph (toward home, not away). Fastest rate away from home is Segment A at 10 mph. (D) is the fastest overall speed but toward home. (C) tempts students who compute the largest distance change without dividing by time correctly. (B) shows no motion.
The temperature in Fahrenheit, F, can be approximated by the equation $F = 2C + 30$ , where C is the temperature in Celsius. Which of the following is the best interpretation of the F-intercept (the vertical intercept) of the graph of this equation?
Answer: A temperature of 0°C is approximately equivalent to 30°F. The F-intercept (y-intercept) is the value of F when C=0. Substituting $C=0$ into the equation gives $F = 2(0) + 30 = 30$ . This means that a temperature of 0 degrees Celsius corresponds to a temperature of 30 degrees Fahrenheit.
The graph below shows two lines, $\ell_1$ and $\ell_2$ . Refer to the figure below. The y-intercept of $\ell_1$ is $k$ times the y-intercept of $\ell_2$ . What is the value of $k$ ?
Answer: $-\tfrac{3}{2}$ From the graph, $\ell_1$ has y-intercept $6$ and $\ell_2$ has y-intercept $-4$ . Then $k=\frac{6}{-4}=-\tfrac{3}{2}$ . (B) inverts the ratio. (C) drops the negative sign and inverts. (D) drops the negative sign.

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