Functions and Graphs

1Summary 重點摘要

Function is like a machine: input x, output y. Drawn on graph paper.

2Vocabulary cards Vocabulary

3Step by step Steps

1. A function is an input-output machineA function works like a machine: feed x in at the entrance, the fixed rule computes, and y leaves at the exit. With y = 2x + 1: input x = 1 and the machine outputs 2 × 1 + 1 = 3; input x = 2 and it outputs 5. The defining feature is that each input has exactly one output — at x = 1, y can only be 3 and never a second value. Water bills and taxi fares are real functions: once the distance is fixed, the fare is fixed too.
2. Coordinates locate pointsEvery point on the plane is a pair (x, y): the first number is the x-coordinate, positive to the right; the second is the y-coordinate, positive upward. Thus (2, 3) sits 2 right and 3 up from the origin (0, 0) where the axes cross. Order matters — (2, 3) and (3, 2) are different points, the classic beginner slip. Remember 'across then up', like a building address giving the floor before the flat number: the order is fixed and must not be swapped.
3. How to draw a straight-line graphTwo points fix a straight line, but plotting three is safer: make a small table first. For y = 2x + 1: x = 0 gives (0, 1); x = 1 gives (1, 3); x = 2 gives (2, 5). Plot the points on graph paper, join them with a ruler and extend both ends. If the third point misses the line, a calculation is wrong and must be re-checked. After drawing, label the line with its equation — only then is the graph complete and properly presented.
4. Gradient m: how steep the line isThe gradient m measures steepness as rise ÷ run: walking along the line, how far you climb per step forward. From (0, 1) to (1, 3): rise = 3 − 1 = 2, run = 1 − 0 = 1, so m = 2 ÷ 1 = 2. Larger m, steeper line; m = 2 means climbing 2 squares for every 1 to the right. When computing, read the two points in the same direction: if y takes 'second minus first', x must too — reversing the order flips the sign and the direction of the line is misjudged.
5. The intercept c and the form y = mx + cThe y-intercept is where the line crosses the y-axis. In the standard form y = mx + c, the letter c is exactly that intercept: y = 2x + 1 crosses at (0, 1), checked by substituting x = 0 to get y = 2 × 0 + 1 = 1. Slope and intercept pin a line down completely — m sets the tilt and c the height — so an equation lets you picture the graph, and a graph lets you read off the equation. This two-way translation between number and shape is the core training of coordinate geometry.
6. The sign of m and the direction of the lineThe sign of the gradient fixes the direction: positive m climbs from lower left to upper right, as in y = 2x + 1; negative m falls from upper left to lower right, as in y = −x + 3; m = 0 is horizontal, as in y = 4. Everyday phrases like 'upward trend' and 'downward trend' are exactly the sign of m. When checking a graph, confirm the direction against the sign of m first and the crossing point against c second; with both right, the line is essentially correct.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Function: input x, output y
• Coordinates: (x, y) horizontal first
• Need at least 2 points to draw line
• Gradient m = rise ÷ run
• y = mx + c: m gradient, c intercept