Summary of Introduction to Linear Functions and Gradient
Introduction to Linear Functions and Gradient: A Student Guide
Introduction
Linear functions describe straight-line relationships between two variables. They are fundamental in algebra and model many real-world situations, from finances to motion. In this guide you will learn how to read, write and use linear functions, and how to find and interpret the gradient (slope) and intercept.
What is a linear function?
A linear function has the form $$y = mx + c$$ where $m$ is the gradient (slope) and $c$ is the y-intercept (constant term). The graph of a linear function is a straight line.
Definition: A linear function is a function whose graph is a straight line and can be written as $y = mx + c$, where $m$ and $c$ are real numbers.
Parts of the equation
- Gradient ($m$): tells how steep the line is. It is the change in $y$ divided by the change in $x$.
- Y-intercept ($c$): the value of $y$ when $x = 0$; the point where the line crosses the $y$-axis.
How to calculate the gradient
The gradient between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $$m = \frac{\text{Change in } y}{\text{Change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$$
Example: For points $(4,5)$ and $(7,20)$ $$m = \frac{20 - 5}{7 - 4} = \frac{15}{3} = 5$$ So the gradient is $5$.
Example: For points $(3,0)$ and $(4,5)$ $$m = \frac{5 - 0}{4 - 3} = \frac{5}{1} = 5$$
Example: For points $(0,-15)$ and $(20,85)$ $$m = \frac{85 - (-15)}{20 - 0} = \frac{100}{20} = 5$$ This shows these points lie on the same line with gradient $5$.
Finding the equation from a gradient and a point
If you know the gradient $m$ and a point $(x_0,y_0)$, substitute into $y = mx + c$ and solve for $c$: $$y_0 = m x_0 + c$$ so $$c = y_0 - m x_0$$
Example: Gradient $m = 5$ and point $(4,5)$ $$c = 5 - 5\cdot 4 = 5 - 20 = -15$$ Equation: $$y = 5x - 15$$
Interpreting gradient and intercept
- If $m > 0$, the line rises left-to-right (positive slope). If $m < 0$, the line falls left-to-right (negative slope). If $m = 0$, the line is horizontal.
- The intercept $c$ gives the starting value when $x = 0$.
Definition: The gradient (or slope) of a line is $m = \frac{y_2 - y_1}{x_2 - x_1}$ and indicates the rate of change of $y$ with respect to $x$.
Table: Comparing related concepts
| Concept | Symbol | What it tells you |
|---|---|---|
| Gradient | $m$ | Change in $y$ per unit change in $x$ |
| Y-intercept | $c$ | Value of $y$ at $x = 0$ |
| Horizontal line | $m = 0$ | No change in $y$ |
| Vertical line | undefined | Not a function in the form $y = mx + c$ |
Practical examples and exercises
- Complete the coordinates for the function $y = 5x - 15$ with given $x$ values:
- $(4,5)$, $(7,20)$, $(0,-15)$, $(20,85)$, $(3,0)$ are all on this line because substituting each $x$ gives the listed $y$.
- Determine the equation of the line through $(0,-15)$ and $(3,0)$:
- Gradient: $$m = \frac{0 - (-15)}{3 - 0} = \frac{15}{3} = 5$$
- Intercept $c = -15$, so $$y = 5x - 15$$
- For the function $y = 6x - 2$, identify:
- Gradient $m = 6$
- Y-intercept $c = -2$
Visual interpretation
- Moving left-to-right:
- On a positive slope you must push up to go along the line.
- On a negative slope you would nosedive (go downward) to follow the line.
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Linear Functions Guide
Klíčové pojmy: Linear functions have the form $y = mx + c$, Gradient $m$ equals $\frac{y_2 - y_1}{x_2 - x_1}$, Y-intercept $c$ is $y$ when $x = 0$, To find $c$ use $c = y_0 - m x_0$, Positive $m$ means line rises left-to-right; negative $m$ means it falls, Points $(x_1,y_1)$, $(x_2,y_2)$ are collinear if gradients between pairs are equal, Check a point lies on a line by substituting into the equation, For $y = 5x - 15$ the points $(4,5)$, $(7,20)$, $(0,-15)$, $(20,85)$, $(3,0)$ lie on the line, For $y = 6x - 2$ gradient is $6$ and intercept is $-2$, Never swap numerator and denominator when computing gradients