≡Mathematics EN 9th grade: Linear function
1. Definition of an Affine Function
An affine function is a function of the type:
\( f(x) = mx + b \)
👉 where:
𝑚 → slope (inclination of the line)
𝑏 → ordinate at the origin
✔ Example:
\( f(x) = 2x + 1 \)
2. Graphical Representation
👉 The graph of an affine function is always a straight line.
Each point has the form:
\( (x, f(x)) \)
✔ Example:
\( f(x) = x + 1 \)
Points:
\( (0,1), (1,2), (2,3) \)
3. Vertical Line
A vertical line has the equation:
\( x = k \)
👉 It does not represent a function, because one value of x has several values of y.
4. Slope and y-intercept
- Slope (m)
Indicates the inclination of the line
𝑚 > 0 → increasing line
𝑚 < 0→ decreasing line
𝑚 = 0 → horizontal line
- ordinate at the origin (b)
It is the value of 𝑦 when:
\( x = 0 \)
✔ Example:
\( f(x) = 2x + 3 \Rightarrow b = 3 \)
5. Determining the slope
The slope between two points is:
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
✔ Example:
\( (1,2) \ \text{e} \ (3,6) \)
\( m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2 \)
6. Equation of a line
General form:
\( y = mx + b \)
✔ Example:
\( y = 3x - 2 \)
7. Equation of a line (given a point and the slope)
Formula:
\( y - y_1 = m(x - x_1) \)
✔ Example:
Point (1,2) and slope 𝑚=3
\( y - 2 = 3(x - 1) \)
\( y = 3x - 1 \)
QUICK TIPS FOR THE EXAM
✔ Linear function → straight line:
\( f(x) = mx + b \)
✔ 𝑚 → slope
✔ 𝑏 → where the line intersects the 𝑦 axis
✔ Slope → use two points
✔ Vertical line → not a function
✔ Equation → know how to convert from point + slope
Did you know?
