SSAT (Secondary School Admission Test) Quantitative Reasoning: Algebra and Functions Syllabus

When solving equations with variables on both sides, what is the first strategic step?

Move all variable terms to one side and all constant terms to the other side by adding/subtracting equal amounts.

What are the four inequality symbols and their meanings?

$<$ less than, $>$ greater than, $\leq$ less than or equal to, $\geq$ greater than or equal to.

What special rule applies when solving inequalities?

When you multiply or divide both sides by a negative number, you must reverse (flip) the inequality sign.

Solve the inequality $-3x < 12$.

$x > -4$ (divide by $-3$ and flip the sign).

On a number line graph of an inequality, when do you use an open circle versus a closed circle?

Open circle for $<$ or $>$ (endpoint not included); closed/filled circle for $\leq$ or $\geq$ (endpoint included).

A number increased by $8$ equals $20$. Write and solve the equation.

$x + 8 = 20$, so $x = 12$.

The sum of three consecutive integers is $48$. Set up an equation.

$n + (n+1) + (n+2) = 48$, which gives $3n + 3 = 48$.

What are the names of the two axes in the coordinate plane?

The horizontal $x$-axis and the vertical $y$-axis.

In the ordered pair $(x, y)$, which coordinate is listed first and what does it tell you?

The $x$-coordinate is first; it tells horizontal distance (right if positive, left if negative) from the origin.

What are the coordinates of the origin?

$(0, 0)$

Name the four quadrants and the sign of $(x, y)$ in each.

QI $(+,+)$, QII $(-,+)$, QIII $(-,-)$, QIV $(+,-)$, numbered counterclockwise starting from upper right.

What is the formula for the slope of a line through points $(x_1, y_1)$ and $(x_2, y_2)$?

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$ (rise over run).

Find the slope of the line through $(1, 2)$ and $(4, 11)$.

$m = \dfrac{11 - 2}{4 - 1} = \dfrac{9}{3} = 3$

What does a positive slope versus a negative slope tell you about a line's direction?

Positive slope rises from left to right; negative slope falls from left to right.

What is the slope of a horizontal line and of a vertical line?

A horizontal line has slope $0$; a vertical line has an undefined slope.

What is the slope-intercept form of a linear equation?

$y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

In $y = -2x + 5$, identify the slope and the $y$-intercept.

Slope $m = -2$; $y$-intercept $b = 5$ (the point $(0, 5)$).

How do you graph a line from slope-intercept form $y = mx + b$?

Plot the $y$-intercept $(0, b)$, then use the slope $m = \frac{\text{rise}}{\text{run}}$ to find a second point, and draw the line.

What is the distance formula between points $(x_1, y_1)$ and $(x_2, y_2)$?

$d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$

Find the distance between $(0, 0)$ and $(3, 4)$.

$d = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5$

What is the midpoint formula between points $(x_1, y_1)$ and $(x_2, y_2)$?

$M = \left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right)$

Find the midpoint of the segment from $(2, 6)$ to $(8, 10)$.

$M = \left( \dfrac{2+8}{2}, \dfrac{6+10}{2} \right) = (5, 8)$

On a line graph, what does an upward-sloping trend indicate about the data?

An increasing trend: the quantity rises as the input (often time) increases.

On a graph, what does a flat (horizontal) segment represent?

No change: the value stays constant over that interval.

What is an arithmetic sequence?

A sequence in which each term is found by adding a fixed number (the common difference) to the previous term.

How do you find the common difference of an arithmetic sequence?

Subtract any term from the term that follows it: $d = a_{n} - a_{n-1}$.

What is the formula for the $n$th term of an arithmetic sequence?

$a_{n} = a_{1} + (n - 1)d$, where $a_1$ is the first term and $d$ is the common difference.

What is a geometric sequence, and how do you find its common ratio?

A sequence where each term is the previous term multiplied by a fixed number (the common ratio). Find it by dividing a term by the previous one: $r = \dfrac{a_{n}}{a_{n-1}}$.

Find the next term in the pattern $2, 6, 18, 54, \ldots$

$162$ (geometric with common ratio $r = 3$).

In a function table with the rule "output $= 2x + 1$," what is the output when the input $x = 5$?

$2(5) + 1 = 11$