This section covers the graphs of various types of functions, including linear, quadratic, cubic, reciprocal, and exponential functions. We will explore their key features, how to sketch them, and how to determine their equations from their graphs.
1. Linear Functions
A linear function has the equation y = mx + c, where m is the gradient and c is the y-intercept.
Gradient (m): The steepness of the line. A positive gradient indicates an increasing line, a negative gradient indicates a decreasing line, and a gradient of zero indicates a horizontal line.
Y-intercept (c): The point where the line crosses the y-axis (the value of y when x = 0).
Sketching: To sketch a linear function, you need to identify two points on the line and draw a straight line through them.
Equation
Gradient (m)
Y-intercept (c)
Shape
$y = 2x + 3$
2
3
Increasing
$y = -x - 1$
-1
-1
Decreasing
$y = 5$
0
5
Horizontal
2. Quadratic Functions
A quadratic function has the equation y = ax2 + bx + c, where a ≠ 0. The graph of a quadratic function is a parabola.
Shape: The parabola opens upwards if a > 0 and downwards if a < 0.
Minimum/Maximum Point: The lowest point (minimum) or highest point (maximum) of the parabola is called the vertex.
Y-intercept: The point where the parabola crosses the y-axis (when x = 0). It is given by (0, c).
X-intercepts: The points where the parabola crosses the x-axis (when y = 0). These are the solutions to the quadratic equation ax2 + bx + c = 0.
Sketching: To sketch a quadratic function, identify the vertex, y-intercept, and x-intercepts.
3. Cubic Functions
A cubic function has the equation y = ax3 + bx2 + cx + d, where a ≠ 0. The graph of a cubic function can have turning points and crosses the y-axis at a point (0, d).
Shape: The shape of the cubic function depends on the value of a. If a > 0, the graph starts low and ends high, and vice versa.
Turning Points: Points where the graph changes direction (local maximum or minimum).
Y-intercept: The point where the graph crosses the y-axis (when x = 0). It is given by (0, d).
Sketching: To sketch a cubic function, identify the y-intercept and any turning points.
4. Reciprocal Functions
A reciprocal function has the equation y = 1/x. Its graph is called a hyperbola.
Asymptote: The graph approaches the y-axis (x = 0) but never actually touches it. This is called a vertical asymptote.
Sketching: To sketch a reciprocal function, consider the asymptotes and the shape of the hyperbola.
5. Exponential Functions
An exponential function has the equation y = abx, where a > 0 and a ≠ 1, and b > 0. The graph of an exponential function increases or decreases rapidly.
Y-intercept: The point where the graph crosses the y-axis (when x = 0). It is given by (0, a).
Asymptote: The graph approaches the x-axis (y = 0) but never actually touches it. This is called a horizontal asymptote.
Sketching: To sketch an exponential function, consider the y-intercept and the asymptote.
Suggested diagram: Graphs of linear, quadratic, cubic, reciprocal, and exponential functions.