PCAT (Pharmacy College Admission Test) Quantitative Reasoning Syllabus

How are the slopes of parallel and perpendicular lines related?

Parallel lines have equal slopes ($m_1 = m_2$). Perpendicular lines have slopes that are negative reciprocals ($m_1 \cdot m_2 = -1$).

What defines a function in terms of inputs and outputs?

A function assigns exactly one output to each input. Equivalently, it passes the vertical line test: no vertical line crosses the graph more than once.

What is the quadratic formula for solving $ax^{2} + bx + c = 0$?

$$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}.$$

What does the discriminant $b^{2} - 4ac$ tell you about a quadratic's roots?

If $b^{2} - 4ac > 0$: two distinct real roots; if $= 0$: one repeated real root; if $< 0$: two complex conjugate roots (no real roots).

What are the coordinates of the vertex of the parabola $y = ax^{2} + bx + c$?

The vertex is at $x = -\frac{b}{2a}$, with $y$ found by substituting that $x$ back in. The parabola opens upward if $a > 0$ and downward if $a < 0$.

State the laws of exponents for products, quotients, and powers of powers.

$x^{m} \cdot x^{n} = x^{m+n}$; $\dfrac{x^{m}}{x^{n}} = x^{m-n}$; $(x^{m})^{n} = x^{mn}$; $x^{0} = 1$ (for $x \neq 0$); $x^{-n} = \dfrac{1}{x^{n}}$.

What is the relationship between exponential and logarithmic forms?

$y = b^{x}$ is equivalent to $x = \log_{b} y$. The logarithm is the inverse of the exponential with the same base.

State the product, quotient, and power rules for logarithms.

$\log_b(MN) = \log_b M + \log_b N$; $\log_b\!\left(\frac{M}{N}\right) = \log_b M - \log_b N$; $\log_b(M^{p}) = p\log_b M$.

What is the change-of-base formula for logarithms?

$$\log_b x = \frac{\log_c x}{\log_c b},$$ commonly written as $\frac{\ln x}{\ln b}$ or $\frac{\log x}{\log b}$.

What is the formula for continuous exponential growth/decay?

$A = A_0 e^{kt}$, where $A_0$ is the initial amount, $k$ is the rate constant ($k>0$ growth, $k<0$ decay), and $t$ is time.

How is the half-life of an exponentially decaying quantity related to its decay constant $k$?

$t_{1/2} = \dfrac{\ln 2}{|k|}$, the time for the quantity to drop to half its value (using $A = A_0 e^{-kt}$).

How do you compute the mean of a data set?

$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i,$$ the sum of all values divided by the number of values $n$.

Define the median and the mode of a data set.

The median is the middle value when data are ordered (the average of the two middle values if $n$ is even). The mode is the value that occurs most frequently.

What is the formula for the sample standard deviation?

$$s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^{2}}.$$ The variance $s^{2}$ is this quantity without the square root.

How is the range and the interquartile range (IQR) of a data set defined?

Range $= \text{max} - \text{min}$. The IQR $= Q_3 - Q_1$, the difference between the third and first quartiles, measuring the spread of the middle 50% of data.

For right-skewed and left-skewed distributions, how do the mean and median compare?

Right-skewed (positive skew): mean > median. Left-skewed (negative skew): mean < median. The mean is pulled toward the longer tail.

What is the classical (theoretical) definition of the probability of an event?

$$P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of equally likely outcomes}},$$ a value between 0 and 1 inclusive.

What is the probability of the complement of an event $E$?

$P(E^{c}) = 1 - P(E)$. The probability that $E$ does not occur.

State the addition rule for the probability of $A$ or $B$.

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$. For mutually exclusive events, $P(A \cap B) = 0$, so $P(A \cup B) = P(A) + P(B)$.

State the multiplication rule for independent events, and the general (conditional) form.

Independent: $P(A \cap B) = P(A)\,P(B)$. General: $P(A \cap B) = P(A)\,P(B \mid A)$, where $P(B \mid A)$ is the conditional probability.

What are the formulas for permutations and combinations of $n$ items taken $r$ at a time?

Permutations (order matters): $P(n,r) = \dfrac{n!}{(n-r)!}$. Combinations (order doesn't matter): $C(n,r) = \dbinom{n}{r} = \dfrac{n!}{r!(n-r)!}$.

What are the key properties of a normal distribution?

It is symmetric and bell-shaped about its mean $\mu$; mean = median = mode; total area under the curve is 1; its spread is set by the standard deviation $\sigma$.

State the empirical (68-95-99.7) rule for a normal distribution.

About 68% of data lie within $\mu \pm 1\sigma$, about 95% within $\mu \pm 2\sigma$, and about 99.7% within $\mu \pm 3\sigma$.

What is a z-score and how is it computed?

A z-score gives how many standard deviations a value is from the mean: $$z = \frac{x - \mu}{\sigma}.$$ It standardizes values to compare across distributions.

What does the Central Limit Theorem state about sample means?

For sufficiently large $n$, the sampling distribution of the sample mean is approximately normal regardless of the population's shape, with mean $\mu$ and standard error $\dfrac{\sigma}{\sqrt{n}}$.

What does a p-value represent in hypothesis testing?

The probability of obtaining results at least as extreme as those observed, assuming the null hypothesis is true. If $p \leq \alpha$ (e.g. 0.05), the null hypothesis is rejected.

State the three primary trigonometric ratios in a right triangle (SOH-CAH-TOA).

$\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$.

State the Pythagorean trigonometric identity.

$\sin^{2}\theta + \cos^{2}\theta = 1$. Dividing by $\cos^{2}\theta$ gives $1 + \tan^{2}\theta = \sec^{2}\theta$.

What is the limit definition of the derivative of $f(x)$?

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.$$ It gives the instantaneous rate of change (slope of the tangent line).

State the power rule for differentiation and the basic derivatives of $e^{x}$ and $\ln x$.

Power rule: $\dfrac{d}{dx}x^{n} = nx^{n-1}$. Also $\dfrac{d}{dx}e^{x} = e^{x}$ and $\dfrac{d}{dx}\ln x = \dfrac{1}{x}$.

State the Fundamental Theorem of Calculus (the evaluation form) and the power rule for integration.

$$\int_a^b f(x)\,dx = F(b) - F(a),\quad \text{where } F'=f.$$ Power rule: $\displaystyle\int x^{n}\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.