Constructing Value Tables and Plotting Linear Functions
To graph a linear function, start by creating a value table. Choose several values of x, substitute each into the function, and calculate the corresponding y values. For example, consider y = x + x - 4. First simplify it to y = 2x - 4. Then, select x values such as -1, 0, 1, 2, and calculate y:
For x = -1: y = 2(-1) - 4 = -2 - 4 = -6; point (-1, -6)
For x = 0: y = 2(0) - 4 = 0 - 4 = -4; point (0, -4) - For x = 1: y = 2(1) - 4 = 2 - 4 = -2; point (1, -2) - For x = 2: y = 2(2) - 4 = 4 - 4 = 0; point (2, 0) Plot these points on graph paper and draw a straight line through them, extending in both directions. This line represents the function y = 2x - 4.
- Simplify the function if needed before creating the table.
- Choose a range of x values including negative, zero, and positive values.
- Calculate y for each x and plot the points accurately.
- Draw a straight line through the points for linear functions.
Understanding the Effect of Parameters in Linear Functions
Consider the family of functions y = 2x + n, where n can be -2, -1, 0, 1, 2, or 3. The slope (2) remains constant, so the steepness and direction of the line do not change. However, changing n shifts the graph vertically:
If n increases, the line moves up parallel to the original line.
If n decreases, the line moves down parallel to the original line. For example, y = 2x + 1 is the line y = 2x shifted up by 1 unit compared to y = 2x. This vertical shift changes the y-intercept but not the slope.
- Slope determines steepness and direction; it remains constant when only n changes.
- Changing n shifts the graph vertically without altering its slope.
- The y-intercept is the value of y when x = 0, equal to n in y = 2x + n.
Graphs of Functions in the Form y = ax and y = ab + c
For y = ax, where a is a rational number, the graph is a straight line passing through the origin (0,0) with slope a. For example, y = (3/2)x passes through (0,0) and rises 3 units for every 2 units moved right.
In functions like y = ab + c, where b is a positive integer, the function is linear with slope a × b and y-intercept c. For example, if y = 2 × 3 + 1, this simplifies to y = 6 + 1 = 7, which is a constant function (a horizontal line). However, if the function is y = a × b × x + c, for example y = 2 × 3 × x + 1 = 6x + 1, the slope is 6 and the y-intercept is 1.
Always identify the slope and intercept to understand the graph's shape and position.
- y = ax graphs pass through the origin with slope a.
- In y = ab + c, identify slope as a × b and intercept as c.
- Check if b is multiplied by x to confirm linearity.
- Plot points using value tables to verify the graph.
Solving Equations Graphically: Intersections and Roots
Graphical methods can solve equations by finding where graphs intersect or where they cross the x-axis (roots). For example, to solve y = 2x + 1 and y = x^2, plot both graphs on the same axes.
The points where the line y = 2x + 1 intersects the curve y = x^2 are the solutions to the equation 2x + 1 = x^2.
To find roots of a function like y = 2x - 4, identify where the graph crosses the x-axis (y=0). Set 2x - 4 = 0, solve algebraically to get x = 2, and confirm by observing the graph crossing at (2,0). This visual method helps estimate solutions and understand their nature.
- Plot both functions on the same graph to find intersections.
- Intersections correspond to solutions of equations.
- Roots are x-values where the graph crosses the x-axis (y=0).
- Use graph to estimate solutions when algebraic methods are complex.
Graphs of Exponential Growth and Decay
Exponential functions model growth and decay processes and have the form y = ab^x, where a > 0 and b > 0.
For exponential growth, b > 1, and the graph curves upwards, increasing rapidly as x increases.
For exponential decay, 0 < b < 1, and the graph curves downwards, approaching the x-axis but never touching it (asymptote). For example, y = 2 × 3^x shows exponential growth, doubling and tripling values as x increases. Conversely, y = 5 × (1/2)^x shows decay, halving values as x increases. These graphs are not straight lines but smooth curves, reflecting the multiplicative rate of change.
- Exponential growth: b > 1, graph rises steeply.
- Exponential decay: 0 < b < 1, graph falls towards x-axis.
- Graphs have a horizontal asymptote (usually y=0).
- Plot points using value tables to capture curve shape.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Create Value Table | Select x values and calculate corresponding y values. | For y=2x-4, x=0 gives y=-4. | Shows accurate plotting of points for graph construction. |
| Plot Points | Mark calculated points on coordinate axes. | Plot (0,-4), (1,-2), (2,0) for y=2x-4. | Ensures correct graph shape and position. |
| Draw Graph | Join points with a straight line for linear functions. | Draw straight line through plotted points. | Demonstrates understanding of linearity and slope. |
| Interpret Parameters | Identify slope and intercept from equation. | In y=2x+1, slope=2, intercept=1. | Explains graph shifts and steepness. |
| Solve Graphically | Find intersections or roots on graph. | Find where y=2x+1 meets y=x^2. | Used to solve equations visually in exams. |
| Recognise Exponential | Identify growth or decay from base b. | y=3^x grows; y=(1/2)^x decays. | Distinguishes curve shapes and real-world models. |
Active recall
Quick check
1. How do you find the slope and y-intercept from y = 3x - 5?
Slope is 3 (coefficient of x), y-intercept is -5 (constant term).
2. What does changing n in y = 2x + n do to the graph?
It shifts the graph vertically up or down without changing the slope.
3. Where does the graph of y = ax always pass through?
The origin (0,0).
4. How can you solve the equation y = 2x + 1 and y = x^2 graphically?
By plotting both graphs and finding their points of intersection.
5. What is the shape of an exponential decay graph?
A curve that decreases and approaches the x-axis but never touches it.