Grade 9 Math Review - Functions and Linear Equations

Identification of Relations and Functions

A critical concept in algebra is distinguishing between a general relation and a function. A relation is defined as a function if, and only if, there are no repeated values for the independent variable, typically represented as the $x$ coordinate. In a set of ordered pairs, each $x$ value must correspond to exactly one $y$ value to satisfy the requirements of a function. For example, in a collection of coordinate pairs, determining if the relation is a function involves scanning the first elements of each pair to ensure they are unique. Based on the review, for a specific set of relations provided in multiple-choice format, option B represents a function because it maintains this unique mapping.

Determining Domain and Range from Ordered Pairs

The domain and range are fundamental properties of any relation or function. The domain refers to the complete set of all possible values for the $x$ coordinate (the independent variable). In the specific example provided, which includes the coordinates $(-2, 4)$, $(0, 1)$, $(3, 4)$, and $(5, 7)$, the domain is identified by extracting the first values of each pair: \text{Domain} = \begin{Bmatrix} -2, 0, 3, 5 \text{ } \bgroup \rgroup. This corresponds to letter C in the review materials. Conversely, the range represents the set of all resulting $y$ values (the dependent variable). For the same set of coordinates, the range is composed of the second values: \text{Range} = \begin{Bmatrix} 4, 1, 4, 7 \bgroup \rgroup. This is identified as letter A.

Mathematical Modeling and Variable Independence

Linear functions are frequently used to model real-world scenarios such as the cost of goods. Consider a situation where the total cost, denoted as $z$, of notebooks is modeled by the equation $z = 35n$. In this specific model, the variable $n$ (representing the number of notebooks) is defined as the independent variable. The total cost is dependent on how many items are purchased; thus, you multiply the unit price by the quantity to find the result. In any such cost function, the variable that you change or control is independent, whereas the resulting total is the dependent variable.

Another example of modeling involves transportation costs. A tricycle ride may cost a base fee of $20$ pesos plus an additional $8$ pesos per kilometer traveled. Using the word "per" as a linguistic indicator for multiplication, the function representing the total cost $C$ for $k$ kilometers is expressed as $C = 8k + 20$ (or $20 + 8x$ where $x$ is kilometers). If one needs to calculate the cost of a delivery service charging $60$ pesos base plus $30$ pesos per kilometer for a $4$ kilometer trip, the function is $f(x) = 60 + 30x$. Substituting the distance, we calculate $60 + 30(4) = 60 + 120$, resulting in a total cost of $180$ pesos.

Structural Characteristics of Linear Equations

A linear function is typically represented in the slope-intercept form, written as $y = mx + b$. In this structure, the variable $m$ represents the slope, which indicates the steepness and direction of the line, while the variable $b$ represents the $y$-intercept, which is the point where the graph crosses the vertical axis. For the equation $y = -3x + 8$, the slope $m$ is $-3$. In the equation $y = 5x - 2$, the slope is $5$ and the $y$-intercept is $-2$. To be classified as a linear function, the highest exponent of the variables must be $1$, and the variable should not appear in the denominator of a fraction.

Graphically, the behavior of a linear function can be predicted by its slope. If the slope is negative, the graph behaves by decreasing as it moves from left to right. This is distinct from a horizontal line (slope of $0$) or a vertical line (undefined slope). When graphing using the slope, the process is often described as "rise over run." For example, if a line starts at the $y$-intercept of $y = 2x + 1$, a slope of $2$ (interpreted as $\frac{2}{1}$) implies a movement of rising $2$ units and running (moving horizontally) $1$ unit to locate the next point.

Calculating Slopes and Intercepts

The slope of a line can be calculated from a table of values or two points using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. Given a table where $x$ values are $0$, $2$, and $4$ and corresponding $y$ values are $1$, $7$, and $13$, we can select two points such as $(0, 1)$ and $(2, 7)$. The calculation becomes $m = \frac{7 - 1}{2 - 0} = \frac{6}{2} = 3$. A linear function is also identifiable in a table if there is a common difference between consecutive $y$ values, provided the $x$ values change at a constant rate. For a table with $y$ values $2$, $5$, $8$, and $11$, the common difference is $3$ ($5-2=3$, $8-5=3$, $11-8=3$), confirming its linear nature.

The "zero" of a function is the value of $x$ that makes the function equal to zero (the $x$-intercept). To find the zero of $f(x) = 2x - 10$, we set the equation to zero: $0 = 2x - 10$. Solving for $x$, we get $-2x = -10$, which yields $x = 5$. To find the $x$-intercept of the equation $y = 2x - 6$, we set $y = 0$: $0 = 2x - 6$. Adding $6$ to both sides results in $6 = 2x$, making $x = 3$. Thus, the $x$-intercept is the point $(3, 0)$.

Comparative Evaluation of Linear Plans and Savings

Linear equations are useful for comparing different services or financial plans. For instance, comparing Plan A ($50x + 100$) and Plan B ($70x$) at a usage level of $x = 5$ reveals that both plans have an identical cost of $350$ ($50 \times 5 + 100 = 350$ and $70 \times 5 = 350$). Similarly, comparing Service X ($40x + 80$) and Service Y ($30x + 150$) at $x = 7$ shows the costs are equal at $360$. Service X results in $40(7) + 80 = 280 + 80 = 360$, while Service Y results in $30(7) + 150 = 210 + 150 = 360$.

In savings scenarios, a linear function can track growth over time. If a savings account starts with a balance of $500$ pesos and increases by $150$ pesos each week, the total saved after $8$ weeks can be modeled by the function $f(x) = 500 + 150x$. Substituting $x = 8$, the calculation is $500 + 150(8) = 500 + 1200 = 1700$ pesos. For students tracking hourly earnings, the domain is often limited to a contextual range, such as integers representing complete hours worked from $1$ to a maximum (e.g., $6$ hours).

Geometric Properties: Parallel and Perpendicular Lines

In geometry, specific conditions define the relationship between non-vertical lines. Two lines are parallel if they have equal slopes. For example, any two lines with a slope of $3$ are considered parallel. Furthermore, parallel lines must be coplanar, meaning they lie in the same plane. This distinguishes them from skew lines, which do not intersect but are not parallel because they are in different planes.

Two lines are perpendicular if their slopes are negative reciprocals of each other. This means the product of their slopes is exactly $-1$. If a line has a slope of $\frac{2}{3}$, a line perpendicular to it must have a slope of $-\frac{3}{2}$. Similarly, if a road has a slope of $\frac{1}{4}$, any crossing road that is perpendicular to it must have a slope of $-4$. These rules ensure that the lines meet at a perfect $90$-degree angle.