AP Calculus AB/BC FRQ Practice

Practice FRQ 1 of 8

1. The following functions are defined for this question:

k(x)=\frac{x^2-4}{x-2}

p(x)=\frac{x^2-4}{x^2-5x+6}

The function

k

is defined for

x\ne 2

. The function

p

is defined for

x\ne 2,3

. Figure 1 shows the graphs of these functions. A graphing calculator may be used to help analyze the behavior of these functions near their points of discontinuity and as

x\to\infty

Figure 1. Graphs of y = k(x) and y = p(x) with discontinuities highlighted (hole at x = 2 for both; vertical asymptote at x = 3 for p).

Single coordinate-plane graph showing BOTH functions on the same axes.

AXES AND SCALE (must be explicit and numeric):
- Horizontal axis labeled "x", drawn as a solid line with an arrowhead on the positive end.
- Vertical axis labeled "y", drawn as a solid line with an arrowhead on the positive end.
- x-axis tick labels shown at every 1 unit from −2 through 6 inclusive: −2, −1, 0, 1, 2, 3, 4, 5, 6.
- y-axis tick labels shown at every 1 unit from −4 through 6 inclusive: −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6.
- The visible plotting window is exactly x from −2 to 6 and y from −4 to 6.
- No grid is required; if grid is drawn, it must align exactly with the integer tick marks.

CURVE 1: y = k(x)
Mathematical identity to enforce the correct shape: k(x) = (x^2 − 4)/(x − 2) simplifies to the straight line y = x + 2 for all x ≠ 2.
- Draw k(x) as a straight, solid line with slope 1 (rises 1 unit for every 1 unit moved right).
- The line must pass exactly through these visible reference intercepts:
• y-intercept at x = 0: the line crosses the y-axis at y = 2.
• x-intercept at y = 0: the line crosses the x-axis at x = −2.
- Mark the removable discontinuity at x = 2 as an OPEN CIRCLE located on the same line at the y-value the line would have there: y = 4.
• The open circle must be centered precisely on the intersection of x = 2 and y = 4.
• The line must be drawn continuously on both sides of x = 2 but with a visible gap at the open circle (do NOT fill it).
- Place a text label "y = k(x)" next to this line in the left half of the graph (for example near x = −1 to 0 region), with a small leader line or positioned so it clearly refers to the line.

CURVE 2: y = p(x)
Mathematical identity to enforce the correct shape: p(x) = (x^2 − 4)/(x^2 − 5x + 6) = (x + 2)/(x − 3) for x ≠ 2,3.
Key required features:
1) Vertical asymptote at x = 3
- Draw a vertical dashed line exactly through the x-axis tick labeled "3".
- Label this dashed line "x = 3" near the top of the plotting window.
- The p(x) curve must approach this line without touching it:
• As x approaches 3 from the left, the curve drops downward without bound (falls toward negative y-values below the window).
• As x approaches 3 from the right, the curve rises upward without bound (rises toward y-values above the window).

2) Removable discontinuity (hole) at x = 2
- Since p(x) matches (x + 2)/(x − 3) wherever defined, the missing point at x = 2 must lie at y = −4.
- Place an OPEN CIRCLE exactly at the intersection of x = 2 and y = −4.
- The curve must approach this open circle smoothly from both sides (no corner), but the point itself is not filled.

3) Horizontal asymptote y = 1 (end behavior)
- Draw a horizontal dashed line at y = 1 across the full width of the window.
- Label this dashed line "y = 1" near the right side of the window.
- The right-hand branch of p(x) (for x > 3) must decrease toward this dashed line from above as x moves right toward 6.
- The left-hand branch of p(x) (for x < 3) must also approach this dashed line as x moves left toward −2, staying near but not identical to it.

4) Exact reference points to lock the curve shape
- Ensure p(x) crosses the x-axis at x = −2 (because the numerator is zero at x = −2 and the denominator is nonzero there).
• The curve must pass through the x-axis exactly at the tick labeled −2 (filled point, because it is defined there).
- Ensure p(x) crosses the y-axis at x = 0 with y = −2/3.
• Plot this crossing below 0 and above −1, closer to −1 than to 0, consistent with −0.666… .
- Ensure p(x) passes through x = 4 with y = 6 (since (4+2)/(4−3) = 6).
• Plot a point on the right branch at x = 4 exactly at y = 6, and draw the curve through it.

5) Overall shape of p(x) by interval (must match rational-function behavior)
- For x from −2 up to just less than 3: draw a continuous branch that
• passes through (x = −2, y = 0),
• crosses the y-axis at y = −2/3,
• has an open circle at (x = 2, y = −4),
• then plunges downward toward negative infinity as it nears the vertical asymptote x = 3 from the left.
- For x from just greater than 3 to 6: draw a second continuous branch that
• starts very high (above the top of the window) near x = 3+,
• goes through the point (x = 4, y = 6),
• then decreases while flattening out and approaching the horizontal asymptote y = 1 by the time it reaches the right edge x = 6.

STYLING AND LABELING (to remove ambiguity):
- Use two clearly distinct styles:
• k(x): solid medium-thick line.
• p(x): solid medium-thick curve in a different color OR a different line style (e.g., solid black for k and solid dark gray for p), but both must remain clearly readable.
- Open circles (holes) must be drawn as unfilled circles with a visible outline.
- The dashed asymptote lines (x = 3 and y = 1) must be thinner than the function graphs.
- Place a text label "y = p(x)" adjacent to the p(x) curve on the right-hand branch (for example near x = 4 to 5, y between 2 and 5) so it clearly refers to that curve.

Everything listed (ticks, labels, holes, asymptotes) must be visible within the window x ∈ [−2, 6], y ∈ [−4, 6].

A. Use correct limit notation to represent

\lim_{x\to 2}k(x)

. Then find the value of the limit. Show the work that leads to your answer.

B. Let

x

approach 2 through the values 1.9, 1.99, 2.01, and 2.1.

(i) Use a calculator to approximate the average rate of change of

k

on the interval

[1.9,2.1]

(ii) Use a calculator to approximate the average rate of change of

k

on the interval

[1.99,2.01]

(iii) Based on your results in parts (i) and (ii), estimate the instantaneous rate of change of

k

x=2

. Justify your reasoning.

C. Write a limit expression that describes the end behavior of

p(x)

x\to\infty

. Then evaluate the limit using limit theorems.

D. Define a new function

q

q(x)=\begin{cases}p(x),& x\ne 2\\ a,& x=2\end{cases}

Find the value of the constant

a

such that

q

is continuous at

x=2

. Justify your answer using the definition of continuity at a point.

Practice FRQ 1 of 8

1. The following functions are defined for this question:

k(x)=\frac{x^2-4}{x-2}

p(x)=\frac{x^2-4}{x^2-5x+6}

The function

k

is defined for

x\ne 2

. The function

p

is defined for

x\ne 2,3

. Figure 1 shows the graphs of these functions. A graphing calculator may be used to help analyze the behavior of these functions near their points of discontinuity and as

x\to\infty

Figure 1. Graphs of y = k(x) and y = p(x) with discontinuities highlighted (hole at x = 2 for both; vertical asymptote at x = 3 for p).

A. Use correct limit notation to represent

\lim_{x\to 2}k(x)

. Then find the value of the limit. Show the work that leads to your answer.

B. Let

x

approach 2 through the values 1.9, 1.99, 2.01, and 2.1.

(i) Use a calculator to approximate the average rate of change of

k

on the interval

[1.9,2.1]

(ii) Use a calculator to approximate the average rate of change of

k

on the interval

[1.99,2.01]

(iii) Based on your results in parts (i) and (ii), estimate the instantaneous rate of change of

k

x=2

. Justify your reasoning.

C. Write a limit expression that describes the end behavior of

p(x)

x\to\infty

. Then evaluate the limit using limit theorems.

D. Define a new function

q

q(x)=\begin{cases}p(x),& x\ne 2\\ a,& x=2\end{cases}

Find the value of the constant

a

such that

q

is continuous at

x=2

. Justify your answer using the definition of continuity at a point.