Edexcel IAL - Pure Maths 2- 5.1 Graph of y = ax- Study notes - New syllabus
Edexcel IAL – Pure Maths 2- 5.1 Graph of y = ax -Study notes- New syllabus
Edexcel IAL – Pure Maths 2- 5.1 Graph of y = ax -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- Graph of y = ax
The Exponential Function \( y = a^x \)
An exponential function is a function in which the variable appears in the exponent.
The general exponential function is:
\( y = a^x \), where \( a > 0 \) and \( a \neq 1 \)
The restriction \( a \neq 1 \) is required because \( 1^x = 1 \) is a constant function, not an exponential curve.
Key Properties of \( y = a^x \)
| Property | Description |
| Domain | All real numbers |
| Range | \( y > 0 \) |
| y-intercept | At \( x = 0 \): \( y = a^0 = 1 \) |
| Asymptote | Horizontal asymptote: \( y = 0 \) |
| Continuity | Graph is smooth and continuous |
Graph Behaviour
Case 1: \( a > 1 \)
- The graph is increasing.
- As \( x \to \infty \), \( y \to \infty \).
- As \( x \to -\infty \), \( y \to 0 \).
Case 2: \( 0 < a < 1 \)
- The graph is decreasing.
- As \( x \to \infty \), \( y \to 0 \).
- As \( x \to -\infty \), \( y \to \infty \).
Important Points on the Graph
- \( (0,1) \) always lies on the graph.
- \( (1,a) \) lies on the graph.
- The graph never touches or crosses the x-axis.
Example
Sketch the graph of \( y = 2^x \).
▶️ Answer / Explanation
- Since \( a = 2 > 1 \), the graph is increasing.
- Passes through \( (0,1) \) and \( (1,2) \).
- Horizontal asymptote is \( y = 0 \).
Example
State the domain, range, and asymptote of the function \( y = \left(\dfrac{1}{3}\right)^x \).
▶️ Answer / Explanation
Here \( 0 < a < 1 \), so the graph is decreasing.
- Domain: all real numbers
- Range: \( y > 0 \)
- Horizontal asymptote: \( y = 0 \)
Example
The graph of \( y = a^x \) passes through the point \( (2,9) \). Find the value of \( a \) and describe the graph.
▶️ Answer / Explanation
Substitute the point into the equation:
\( 9 = a^2 \)
\( a = 3 \) (since \( a > 0 \))
The function is:
\( y = 3^x \)
Graph description:
- Increasing exponential curve
- Passes through \( (0,1) \) and \( (2,9) \)
- Horizontal asymptote: \( y = 0 \)