Tangent Gradient at y-Intercept (HSC SSCE Mathematics Advanced): Revision Notes

Tangent Gradient at y-Intercept

Introduction

By the end of this section, you should be able to:

• Define what a tangent and a secant are in relation to a curve
• Understand how the gradient of a secant line can be used to approximate the gradient of a tangent
• Use the formula $m = \frac{a^h - 1}{h}$ to estimate the gradient of the tangent to $y = a^x$ at its y-intercept
• Examine how the gradient of the tangent at the y-intercept varies depending on the value of the base $a$

Key terminology

Understanding these fundamental terms is essential for working with tangent gradients.

Tangent

A tangent is a straight line that touches a curve at exactly one point. When we talk about the tangent to a curve at a specific point, we mean a line that just touches the curve there, with a gradient that matches the curve's rate of change at that exact location.

Gradient

The gradient is the slope or steepness of a line. If you have two distinct points on a line, $A(x_1, y_1)$ and $B(x_2, y_2)$, you can calculate the gradient using:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

This formula tells you how much the line rises (or falls) for each unit you move horizontally.

Intercept

An intercept is where a curve or function crosses an axis. The x-intercept occurs where the curve crosses the x-axis (when $y = 0$), and the y-intercept occurs where the curve crosses the y-axis (when $x = 0$).

Secant

A secant is a straight line that passes through two points on a curve. Unlike a tangent which touches at one point, a secant cuts through the curve at two different points.

Understanding the tangent gradient at the y-intercept

The tangent to a curve at any point is a line that touches the curve at that specific point. The gradient of this tangent line equals the curve's instantaneous rate of change at that point.

For an exponential function $y = a^x$ (where $a > 0$ and $a \neq 1$), finding the y-intercept is straightforward. The y-intercept occurs when $x = 0$:

$y = a^0 = 1$

Estimating the gradient using a secant line

To find the gradient of the tangent at $x = 0$, we can approximate it using a secant line. This method works by connecting two points on the curve: the y-intercept $(0, 1)$ and a nearby point $(h, a^h)$, where $h$ is a small number.

The gradient of this secant line is calculated as:

$m = \frac{a^h - 1}{h}$

Here's what each part means:

• $m$ represents the gradient of the secant line
• $a^h$ is the y-value at $x = h$
• $1$ is the y-value at $x = 0$
• $h$ is the horizontal distance between the two points

This formula allows you to estimate the rate of change of $y = a^x$ at $x = 0$. By testing different values of $a$, you can observe how the gradient changes for different exponential functions. For instance:

• When $a = 2$, the gradient is less than 1
• When $a = 3$, the gradient is greater than 1

Exploration activity

To develop your understanding of how this formula works, try the following investigation:

Calculate the gradient of the secant for $y = 2^x$ at $x = 0$ using $h = 0.1$ and then $h = 0.01$. Then repeat this process for $y = 3^x$. Notice how the gradient changes as you vary $a$, and observe what happens as $h$ gets progressively smaller.

Worked example

Let's work through a complete example to see how to apply the secant gradient formula.

Key observation: The gradient of the tangent to $y = a^x$ at the y-intercept $(0, 1)$ varies with the value of $a$. This gradient can be estimated using secant lines, with the approximation improving as $h$ gets smaller.

Remember!