This question tests your understanding of average rate of change, which is a super important concept connecting slope, functions, and real-world rates like speed or growth. The average rate of change has units that come from dividing output units by input units: if distance is in miles and time is in hours, the rate is in miles per hour (mi/hr). To find the average rate of change of d(t) = 6t + 2 from t = 1 to t = 4, we first evaluate at the endpoints: d(1) = 61 + 2 = 8 and d(4) = 64 + 2 = 26. Then we use the formula: average rate = [d(4) - d(1)]/(4 - 1) = (26 - 8)/3 = 18/3 = 6 mi/hr. Choice A is correct because it properly calculates [d(4) - d(1)]/(4 - 1) = 18/3 = 6 mi/hr, getting both the arithmetic and the units right! Choice C calculates the change in y correctly as 18, but forgets to divide by the change in t (which is 3). Units are your friend for understanding: if distance is in feet and time is in seconds, then average rate of change is in feet per second (ft/sec). The 'per' in the units reminds you that it's a ratio—change in output PER change in input. This helps you interpret what the number means!

This question tests your understanding of average rate of change, which is a super important concept connecting slope, functions, and real-world rates like speed or growth. For a linear function, the average rate of change is the same as the slope and doesn't depend on which interval you choose—the function changes at a constant rate everywhere. But for nonlinear functions like quadratics, the average rate of change can be different over different intervals. To find the average rate of change of p(x)=4x-7 from x=-2 to x=3, we first evaluate at the endpoints: p(-2)=4*(-2)-7=-15 and p(3)=4*3-7=5. Then we use the formula: average rate = [5 - (-15)]/(3 - (-2)) = 20/5 = 4. That's it! Choice B is correct because it properly calculates [p(3) - p(-2)]/(3 - (-2)) = 20/5 = 4, getting both the arithmetic and the sign right! Choice A flips the order in the numerator or denominator, calculating [p(-2) - p(3)]/(3 - (-2)) = -20/5 = -4, which changes the sign of the answer. The formula is always (later y - earlier y)/(later x - earlier x), maintaining consistent order! Here's a way to remember: average rate of change is just slope between two points. If you can find slope, you can find average rate of change—it's the same calculation! For linear functions, this slope is constant. For curves, the slope of the secant line connecting two points gives you the average rate over that interval.

This question tests your understanding of average rate of change, which is a super important concept connecting slope, functions, and real-world rates like speed or growth. For a linear function, the average rate of change is the same as the slope and doesn't depend on which interval you choose—the function changes at a constant rate everywhere. But for nonlinear functions like quadratics, the average rate of change can be different over different intervals. To find the average rate of change of d(t) = 50t + 20 from t = 1 to t = 4, we first evaluate at the endpoints: d(1) = 50(1) + 20 = 50 + 20 = 70 and d(4) = 50(4) + 20 = 200 + 20 = 220. Then we use the formula: average rate = [d(4) - d(1)]/(4 - 1) = (220 - 70)/(4 - 1) = 150/3 = 50. That's it! Choice B is correct because it properly calculates [d(4) - d(1)]/(4 - 1) = 150/3 = 50 mi/hr, getting both the arithmetic and the units right! Choice C gives just 200 miles, which is d(4) - 20, not the rate of change. Remember: average rate of change = (change in y)/(change in x), not just one or the other. We're finding how much distance changes per unit of time! Units are your friend for understanding: if distance is in miles and time is in hours, then average rate of change is in miles per hour (mi/hr). The 'per' in the units reminds you that it's a ratio—change in output PER change in input. This helps you interpret what the number means!

This question tests your understanding of average rate of change, which is a super important concept connecting slope, functions, and real-world rates like speed or growth. For a linear function, the average rate of change is the same as the slope and doesn't depend on which interval you choose—the function changes at a constant rate everywhere. But for nonlinear functions like quadratics, the average rate of change can be different over different intervals. To find the average rate of change of m(x)=(1/2)x+3 from x=4 to x=10, we first evaluate at the endpoints: m(4)=(1/2)*4+3=2+3=5 and m(10)=(1/2)*10+3=5+3=8. Then we use the formula: average rate = [8 - 5]/(10 - 4) = 3/6 = 1/2. That's it! Choice B is correct because it properly calculates [m(10) - m(4)]/(10 - 4) = 3/6 = 1/2, getting both the arithmetic and the sign right! Choice D has the right idea but makes an arithmetic error: it calculates 3/ (something wrong) or confuses with total change; these calculations can be tricky, especially with fractions—always good to double-check! Here's a way to remember: average rate of change is just slope between two points. If you can find slope, you can find average rate of change—it's the same calculation! For linear functions, this slope is constant. For curves, the slope of the secant line connecting two points gives you the average rate over that interval.

This question tests your understanding of average rate of change, which is a super important concept connecting slope, functions, and real-world rates like speed or growth. The average rate of change has units that come from dividing output units by input units: if value is in dollars and time is in years, the rate is in dollars per year ($/year). To find the average rate of change of V(t) = 200 - 15t from t = 2 to t = 8, we first evaluate at the endpoints: V(2) = 200 - 152 = 170 and V(8) = 200 - 158 = 80. Then we use the formula: average rate = [V(8) - V(2)]/(8 - 2) = (80 - 170)/6 = -90/6 = -15 $/year. Choice A is correct because it properly calculates [V(8) - V(2)]/(8 - 2) = -90/6 = -15 $/year, getting both the arithmetic and the units right! Choice B has the magnitude right but the wrong sign. When V(8) is less than V(2), the numerator is negative, giving a negative rate, not positive. Units are your friend for understanding: the 'per' in the units reminds you that it's a ratio—change in output PER change in input. This helps you interpret what the number means!

This question tests your understanding of average rate of change, which is a super important concept connecting slope, functions, and real-world rates like speed or growth. The average rate of change has units that come from dividing output units by input units: if cost is in dollars and distance is in miles, the rate is in dollars per mile ($/mile). To find the average rate of change of C(x) = 3 + 2x from x = 2 to x = 7, we first evaluate at the endpoints: C(2) = 3 + 22 = 7 and C(7) = 3 + 27 = 17. Then we use the formula: average rate = [C(7) - C(2)]/(7 - 2) = (17 - 7)/5 = 10/5 = 2 $/mile. Choice B is correct because it properly calculates [C(7) - C(2)]/(7 - 2) = 10/5 = 2 $/mile, getting both the arithmetic and the units right! Choice C calculates the change in y correctly as 10, but forgets to divide by the change in x (which is 5). Units are your friend for understanding: the 'per' in the units reminds you that it's a ratio—change in output PER change in input. This helps you interpret what the number means!

This question tests your understanding of average rate of change, which is a super important concept connecting slope, functions, and real-world rates like speed or growth. Average rate of change tells us how fast a function is changing on average over an interval: it's calculated as (change in y)/(change in x), or [f(b) - f(a)]/(b - a) when going from x = a to x = b. Think of it as the slope of the line connecting two points on the function's graph! To find the average rate of change of f(x)=2x²+1 from x=1 to x=3, we first evaluate at the endpoints: f(1)=2(1)²+1=3 and f(3)=2(9)+1=19. Then we use the formula: average rate = [19 - 3]/(3 - 1) = 16/2 = 8. That's it! Choice B is correct because it properly calculates [f(3) - f(1)]/(3 - 1) = 16/2 = 8, getting both the arithmetic and the sign right! Choice D flips the order in the numerator or denominator, calculating [f(1) - f(3)]/(3 - 1) = -16/2 = -8, which changes the sign of the answer. The formula is always (later y - earlier y)/(later x - earlier x), maintaining consistent order! The key formula for average rate of change is (y₂ - y₁)/(x₂ - x₁), which you might recognize as the slope formula! To use it: (1) identify your two points or endpoints, (2) subtract the y-values (later minus earlier), (3) subtract the x-values (later minus earlier), (4) divide. Keep the order consistent and you'll get the right answer every time!

This question tests your understanding of average rate of change, which is a super important concept connecting slope, functions, and real-world rates like speed or growth. The average rate of change has units that come from dividing output units by input units: if value is in dollars and time is in years, the rate is in dollars per year ($/year). These units help us understand what the number means—it's not just abstract math! To find the average rate of change of V(t) = 900 - 80t from t = 2 to t = 5, we first evaluate at the endpoints: V(2) = 900 - 80(2) = 900 - 160 = 740 and V(5) = 900 - 80(5) = 900 - 400 = 500. Then we use the formula: average rate = [V(5) - V(2)]/(5 - 2) = (500 - 740)/(5 - 2) = -240/3 = -80. That's it! Choice B is correct because it properly calculates [V(5) - V(2)]/(5 - 2) = -240/3 = -80 $/year, getting both the arithmetic and the interpretation right! Choice C gives just the change in value (-240 dollars) instead of the rate. Remember: average rate of change = (change in y)/(change in x), not just one or the other. We're finding how much value changes per unit of time! The average rate of change of -80 $/year means that on average, the laptop value is decreasing by $80 for each year. In this context, that translates to the laptop depreciating at a constant rate of $80 per year—the negative sign tells us it's losing value, not gaining!

This question tests your understanding of average rate of change, which is a super important concept connecting slope, functions, and real-world rates like speed or growth. But for nonlinear functions like quadratics, the average rate of change can be different over different intervals. Over interval [0,2], the rate is [h(2) - h(0)]/(2 - 0) = (4 - 0)/2 = 2. Over interval [2,4], the rate is [h(4) - h(2)]/(4 - 2) = (16 - 4)/2 = 6. Comparing these: 6 > 2, so interval [2,4] has the larger average rate of change. This makes sense because the function is steeper there. Choice C is correct because it properly calculates and compares the rates to identify [2,4] as larger, getting the interpretation right! Choice A states the interpretation backwards: [0,2] has the smaller rate, not larger. Quick sanity check: if the function is increasing (going up) on your interval, the average rate should be positive. If decreasing (going down), it should be negative. If horizontal, it should be zero. Does your answer match what you see happening in the function? If not, recheck your work!

This question tests your understanding of average rate of change, which is a super important concept connecting slope, functions, and real-world rates like speed or growth. For a linear function, the average rate of change is the same as the slope and doesn't depend on which interval you choose—the function changes at a constant rate everywhere. To find the average rate of change of p(x) = -3x + 4 from x = -2 to x = 2, we first evaluate at the endpoints: p(-2) = -3*(-2) + 4 = 10 and p(2) = -3*2 + 4 = -2. Then we use the formula: average rate = [p(2) - p(-2)]/(2 - (-2)) = (-2 - 10)/4 = -12/4 = -3. Choice A is correct because it properly calculates [p(2) - p(-2)]/(2 - (-2)) = -12/4 = -3, getting both the arithmetic and the sign right! Choice B has the magnitude right but the wrong sign. When p(2) is less than p(-2), the numerator is negative, giving a negative rate, not positive. The key formula for average rate of change is (y₂ - y₁)/(x₂ - x₁), which you might recognize as the slope formula! To use it: (1) identify your two points or endpoints, (2) subtract the y-values (later minus earlier), (3) subtract the x-values (later minus earlier), (4) divide. Keep the order consistent and you'll get the right answer every time!