Math 3 Quiz: Connecting Representations

What this quiz covers

This quiz focuses on Connecting Representations, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 3.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Apply Vieta's formulas to the quartic polynomial to establish relationships between all four zeros, then use the known zeros to determine the unknown ones
2. Use the graph to estimate the locations of the other two zeros visually, then verify these estimated values satisfy the original polynomial equation
3. Since $P(x) = (x-2)(x-4)(x^2 + ax + b)$, divide the original polynomial by $(x-2)(x-4)$ to find the quadratic, then analyze its discriminant
4. Expand $(x-2)(x-4) = x^2 - 6x + 8$, perform polynomial long division to find the remaining quadratic factor, then solve using the quadratic formula (correct answer)

Question 2

1. Confirm that $g(\frac{\pi}{8}) = 3\sin(0) + 1 = 1$ matches the graph, and verify the range spans from $-2$ to $4$ as predicted
2. Verify the amplitude by measuring peak-to-center distance, confirm period by finding cycle completion points, and locate phase shift using the first maximum
3. Use the standard form parameters: amplitude = |3| = 3, period = $\frac{2\pi}{2} = \pi$, phase shift = $\frac{\pi/4}{2} = \frac{\pi}{8}$, vertical shift = 1
4. Check that the function oscillates between $y = -2$ and $y = 4$, completes two full cycles in $[0, 2\pi]$, and starts its cycle at $\theta = \frac{\pi}{8}$ (correct answer)

Question 3

1. Calculate the decay rate from the base 0.8, then use the graph to verify this rate matches the visual slope pattern
2. Use the graph to estimate where $N(t) = 250$, then substitute that t-value back into the original function algebraically
3. Solve the equation $250 = 500 \cdot (0.8)^t$ using logarithms, then locate the point $(t, 250)$ on the graph to confirm (correct answer)
4. Find the intersection of $y = 250$ and $y = 500 \cdot (0.8)^t$ graphically, then verify algebraically using exponential properties

Question 4

1. Calculate $\lim_{x \to 2^-} h(x) = 0$ and $\lim_{x \to 2^+} h(x) = 0$, showing the limits are equal, contradicting the jump discontinuity claim
2. Evaluate $\lim_{x \to 2^-} h(x) = 0$, $\lim_{x \to 2^+} h(x) = 0$, and $h(2) = 0$, demonstrating the function is actually continuous at $x = 2$ (correct answer)
3. The graph shows both pieces approach the same y-value at $x = 2$, and algebraically both one-sided limits equal 0, contradicting the discontinuity claim
4. Since $h(2) = 0$ from the linear piece and both limits approach 0, the function has a removable discontinuity, not a jump discontinuity

Question 5

1. Checking that the vertex form gives $f(1) = 4$ and confirming the vertex location matches the given properties (correct answer)
2. Expanding the vertex form to standard form and comparing coefficients with the original function's requirements
3. Verifying that the axis of symmetry formula $x = -\frac{b}{2a}$ equals 2 for both representations of the function
4. Confirming that both forms have the same discriminant value and therefore the same number of real roots

Question 6

1. The algebraic solution satisfies both original equations, and the geometric intersection point has the same coordinates, providing independent verification methods (correct answer)
2. The substitution method eliminates variables systematically, while the graph visually confirms that exactly one solution exists for this system
3. Both representations show the same point $(3, 2)$, and checking this point in both original equations confirms it satisfies the system requirements
4. The graph demonstrates that the lines have different slopes, ensuring intersection, while algebra provides the precise coordinates of that intersection point

Question 7

1. The empirical rule states 95% falls within $\mu \pm 2\sigma = [34, 66]$, and z-scores of -2 and +2 correspond to the same interval boundaries
2. Calculate $P(-2 < Z < 2) = 0.9544$ using standard normal tables, which matches the empirical rule's 95% prediction for the interval $[34, 66]$ (correct answer)
3. The interval $[50 - 2(8), 50 + 2(8)] = [34, 66]$ contains 95% by the empirical rule, and converting these boundaries to z-scores gives $\pm 2$
4. Both methods identify the same critical interval boundaries, and the slight difference between 95.44% and 95% demonstrates the empirical rule's approximation accuracy

Question 8

1. The expression $(x + 3)$ requires $x > -3$ for the logarithm to be defined, giving domain $(-3, \infty)$, and the range remains $(-\infty, \infty)$ for all logarithmic functions
2. The horizontal shift creates a vertical asymptote at $x = -3$ which bounds the domain, while the vertical shift translates the entire range down by 1 unit
3. From $x + 3 > 0$, the domain is $x > -3$, and since logarithms can output any real number, the range is all real numbers, confirmed by the graph's behavior (correct answer)
4. The asymptote at $x = -3$ shows where the function is undefined, establishing the domain boundary, and the vertical shift affects the y-intercept but not the range

Question 9

1. The positive slope in the regression equation combined with the strong positive correlation coefficient demonstrates clear causal relationship (correct answer)
2. The scatter plot shows an upward trend that aligns with the positive regression slope, providing visual confirmation of causation
3. The high correlation value explains 61% of the variance, which is sufficient statistical evidence to establish causal mechanisms
4. The y-intercept of 65 represents the baseline score without studying, proving that study time directly influences performance

Question 10

1. All three representations are consistent and correctly support the claim about the minimum value and location (correct answer)
2. The completed square form is incorrect; it should be $f(x) = 2(x - 2)^2 + 2$, making the minimum value $2$
3. The vertex coordinates are wrong; the correct vertex is at $(2, 2)$ based on the given algebraic form
4. The algebraic form contradicts the other representations; substituting $x = 2$ gives $f(2) = 2$ not $-2$

Question 11

1. Use the algebraic solution $x = \frac{\pi}{12} + \pi k$ to generate all solutions, then verify these appear correctly in both the table and graph
2. Check that the table shows $\sin(2x) = \frac{1}{2}$ at exactly 4 values, confirm these on the graph where the curve intersects $y = \frac{1}{2}$, and verify algebraically using $2x = \frac{\pi}{6} + 2\pi k$ and $2x = \frac{5\pi}{6} + 2\pi k$ (correct answer)
3. Count intersection points on the graph, use the table to estimate these values more precisely, then solve $\sin(2x) = \frac{1}{2}$ to confirm exactly
4. Recognize that $\sin(u) = \frac{1}{2}$ when $u = \frac{\pi}{6}$ or $\frac{5\pi}{6}$ in $[0, 2\pi]$, then substitute $u = 2x$ and solve for $x$ values in the given interval

Question 12

1. The contradiction indicates an error; the graph should show a vertical asymptote at $x = 3$ since the function is undefined there
2. The function has a removable discontinuity at $x = 3$; simplifying gives $f(x) = x + 3$ for $x \neq 3$, so the graph shows $y = x + 3$ with a hole at $(3, 6)$ (correct answer)
3. The table and graph are incorrect; since $f(3)$ is undefined, the function cannot have any values near $x = 3$
4. The function simplifies to $f(x) = x + 3$ everywhere, including at $x = 3$, so there is no contradiction between the representations

Question 13

1. Find the vertex of the parabola $y = -\frac{2x^2}{25} + \frac{8x}{5}$ using $x = -\frac{b}{2a} = 20$, confirm this matches $x(1) = 20$, and verify $y(1) = 16$ is the maximum height
2. Take the derivative $\frac{dy}{dt} = -32t + 32$, set equal to zero to find $t = 1$, then verify this gives $x = 20$ and represents the maximum height using the second derivative test
3. The claim is incorrect; the maximum occurs when $\frac{dx}{dt} = 0$, which never happens since $\frac{dx}{dt} = 20$ is constant, indicating uniform horizontal motion
4. Check the table values around $t = 1$ to confirm the height is maximized, verify the Cartesian equation gives the same maximum at $x = 20$, and use calculus on $y(t)$ to confirm $t = 1$ is correct (correct answer)

Question 14

1. The claim is correct; the coefficient -0.1 in the linearized equation directly represents the decay constant, and this value makes the exponential model $N(t) = 500e^{-0.1t}$ valid
2. The semi-log plot should show a straight line with slope -0.1, the half-life formula $t_{1/2} = \frac{\ln(2)}{k}$ gives $t_{1/2} = \frac{\ln(2)}{0.1} \approx 6.93$ hours, confirming the decay constant is 0.1 per hour
3. The claim is incorrect; the decay constant should be $\frac{\ln(2)}{6.93} \approx 0.1$, but this approximation shows the true value isn't exactly 0.1 per hour
4. The claim is imprecise; while the decay constant is 0.1 per hour, saying it's 'exactly $\frac{1}{10}$' requires verification that this wasn't rounded from experimental data or parameter estimation (correct answer)

Question 15

1. The data table shows that reaction times for 7-8 hours of sleep are consistently lower than the model predicts, while times for 4-5 hours are consistently higher than predicted
2. The correlation coefficient of $-0.78$ indicates that only 61% of the variation in reaction time is explained by hours of sleep
3. The residual plot displays a clear U-shaped pattern, with systematic positive residuals at both low and high sleep values and negative residuals at moderate sleep values (correct answer)
4. The y-intercept of 250 milliseconds represents an unrealistic reaction time for someone with zero hours of sleep, suggesting the model fails for extreme values