MATH 1 EOC - MUST KNOWS

Linear Functions The slope, represented by the variable $m$, describes the constant rate of change in a linear function and is calculated using two points $(x_1, y_1)$ and $(x_2, y_2)$ through the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. This is conceptually defined as the change in y ($\Delta y$) over the change in x ($\Delta x$). In a linear function, the rate of change must be constant to qualify as linear. The slope-intercept form is given by $y = mx + b$, where $m = \text{slope}$ and $b = \text{y-intercept}$. Visualizing the slope reveals four types: positive (increasing), negative (decreasing), zero (horizontal), and undefined (vertical). In tabular form, a linear function follows an addition pattern: for x-values $0, 1, 2, 3$, the y-values are $a$, $a+d$, $a+2d$, and $a+3d$. Students must be able to find the slope, write equations based on data, and compare different rates. # Exponential Functions Exponential functions follow the general form $y = a(b)^x$, where the base $b > 1$ indicates growth and $0 < b < 1$ indicates decay. Unlike linear functions, exponential functions are characterized by a multiply pattern or constant ratio. Specific growth and decay calculations use the formulas $y = a(1+r)^x$ for growth and $y = a(1-r)^x$ for decay, where $r$ is the rate. In a table of values where x increases sequentially from $0, 1, 2, 3$, the y-values demonstrate the constant ratio: $a$, $a \cdot b$, $a \cdot b^2$, and $a \cdot b^3$. # Sequences Sequences are ordered sets of numbers following specific rules. Arithmetic sequences involve a constant addition pattern ($d = \text{common difference}$) defined by the explicit formula $a_n = a_1 + (n-1)d$. For example, in the sequence $3, 7, 11, 15, \dots$, the difference is $+4$. Geometric sequences involve a multiplication pattern ($r = \text{common ratio}$) defined by the explicit formula $a_n = a_1 \cdot r^{(n-1)}$. For example, in the sequence $2, 6, 18, 54, \dots$, the ratio is $\times 3$. # Solving Equations To solve equations, distribute any values outside parentheses, combine like terms, and then isolate the variable. There are three types of outcomes. A single solution occurs when the variable is equal to a specific value, such as in the example $2x + 3 = 11 \rightarrow 2x = 8 \rightarrow x = 4$. NO SOLUTION occurs when variables cancel and result in a false statement like $3 = 5$, which graphically corresponds to parallel lines. INFINITE SOLUTIONS occur when variables cancel and result in a true statement like $4 = 4$, which graphically corresponds to the same line. # Functions A function is a relation where every one input has exactly one output. In a table, this means there are no repeated x-values paired with different y-values. Graphically, a relation is a function if it passes the Vertical Line Test. In a mapping diagram, every input must have exactly one arrow pointing to an output. # Inequalities Inequalities are solved like equations, with the critical exception that you must flip the inequality sign if you multiply or divide by a negative number. For example, if given $-2x + 6 \le 10$, you subtract 6 to get $-2x \le 4$, then divide by $-2$ and flip the sign to get $x \ge -2$. When graphing on a number line, use an open circle for $<$ or $>$ and a closed circle for $\le$ or $\ge$. # Systems of Equations The solution to a system of equations is the intersection point of the lines. There are three possible scenarios: one solution (lines cross), no solution (lines are parallel), or infinite solutions (the equations describe the same line). Systems can be solved using the graphing, substitution, or elimination methods. # Quadratics Quadratic functions are written in the form $y = ax^2 + bx + c$. Features of the parabola include the vertex (which is either a maximum or minimum), the axis of symmetry, and the zeros (the x-intercepts). If $a > 0$, the parabola opens up; if $a < 0$, it opens down. Typical factoring methods for quadratics include finding the Greatest Common Factor (GCF), factoring trinomials, and recognizing the difference of squares. # Statistics The mean is the average calculated as the sum of data divided by the number of data points. The median is the middle value, or the average of the two middle values if the data set is even. The range is the difference between the maximum and minimum values ($max - min$). The Interquartile Range ($IQR$) is calculated as $Q_3 - Q_1$. Greater spread in the data indicates higher variability. # Geometry The Pythagorean Theorem, $a^2 + b^2 = c^2$, is used to find missing side lengths in right triangles only. # Scatter Plots Scatter plots show correlations between data: positive correlation (upward trend), negative correlation (downward trend), or none (random distribution). A line of best fit is used as a tool for making predictions. # Exponent Rules There are three primary rules for exponents. The Product Rule states that for $x^m \cdot x^n$, you add exponents to get $x^{m+n}$. The Quotient Rule states that for $\frac{x^m}{x^n}$, you subtract exponents to get $x^{m-n}$. The Power to a Power Rule states that for $(x^m)^n$, you multiply exponents to get $x^{mn}$. # Test Tips Critical strategies for the exam include identifying whether a function is linear or exponential first, watching for repeated x-values that would disqualify a relation as a function, remembering to flip inequalities when dealing with negatives, using graphs as a visual aid when stuck, and labeling all answers clearly.