Grade 9 Mathematics: Linear and Non-Linear Graphs

Linear Graphs: Represented by a constant rate of change and a straight-line pattern. The exponent of the unknown variable is always $1$ .
Non-Linear Graphs: Occur when the rate of change is not constant, resulting in a curved pattern or a change in direction. The exponent of the unknown variable is greater than $1$ .

Constant Function: Occurs when the $y$ -axis variable remains unchanged while the $x$ -axis variable changes; the graph is parallel to an axis.
Increasing Function: Indicates a direct proportion where the $y$ -axis value increases as the $x$ -axis value increases. The graph moves upwards from left to right.
Decreasing Function: Indicates an indirect proportion where the $y$ -axis value decreases as the $x$ -axis value increases. The graph moves downwards from left to right.

Maximum Value: The highest point or value on the graph relative to the $y$ -axis.
Minimum Value: The lowest point or value on the graph relative to the $y$ -axis.
Discrete Data: Data that is counted and involves only whole numbers. Points on these graphs remain as unconnected dots.
Continuous Data: Data that is measured and involves real numbers. Points are connected to form a solid line or curve because all intermediate values are possible.

x-intercept: The point where the graph cuts the $x$ -axis; calculated by substituting $y = 0$ into the equation.
y-intercept: The point where the graph cuts the $y$ -axis; calculated by substituting $x = 0$ into the equation.
Gradient ( $m$ ): The measure of slope or steepness, defined as the rate at which $y$ -values change relative to $x$ -values.
Gradient Formula: $m = \frac{\text{change in y-values}}{\text{change in x-values}}$ or $m = \frac{y_b - y_a}{x_b - x_a}$ .
Gradient Properties: A positive gradient indicates an increasing function, while a negative gradient indicates a decreasing function. A larger absolute value indicates a steeper slope.

Structure: A system where points are described by ordered pairs $(x; y)$ . The horizontal line is the $x$ -axis and the vertical line is the $y$ -axis.
Origin: The intersection of the axes where both have a value of $0$ .
Independent Variable: Placed on the $x$ -axis; it can be changed without effect from other variables.
Dependent Variable: Placed on the $y$ -axis; its value depends on the changes made to the independent variable.

Standard Equation: The standard form of a linear graph is $y = mx + c$ , where $m$ is the gradient and $c$ is the $y$ -intercept.
Dual Intercept Method: A technique for drawing graphs by determining and plotting the $x$ -intercept and $y$ -intercept, then joining them with a straight line.
Vertical Shifting: Changing the value of $c$ ( $y$ -intercept) shifts the graph vertically. Increasing $c$ moves the graph upwards; decreasing $c$ moves it downwards. These shifts result in parallel lines as the gradient remains unchanged.
Determining Equations: To find the equation from a graph or table, identify the constant difference ( $m$ ) and the $y$ -intercept ( $c$ ) where $x = 0$ . If the $y$ -intercept is not given, solve for $c$ by substituting a known ordered pair into $y = mx + c$ .