Slopes of Lines and Curves

The Slope of a Line

Linear functions produce straight-line graphs. In general, a straight line follows the following equation:

$y = a + bx$

where $a$ and $b$ are fixed numbers.

The line on the graph is the set of points:

$\left[ (x,y): x,y \in \mathbb{R}, y = a+bx \right]$

Details

The slope of a straight line represents the change in the $y$ coordinate corresponding to a unit change in the $x$ coordinate.

Segment Slopes

Let's assume we have a more general function $y = f(x)$. To find the slope of a line segment, consider two $x$ -coordinates ($x_0$ and $x_1$), and look at the slope between $(x_0, f(x_0))$ and $(x_1, f(x_1))$.

Details

Consider two points, $(x_0,y_0)$ and $(x_1,y_1)$. The slope of the straight line that goes through these points is

$\displaystyle\frac {y_1 - y_0} {x_1 - x_0}$

Thus, the slope of a line segment passing through the points $(x_0,f(x_0))$ and $(x_1,f(x_1))$, for some function, $f$, is

$\displaystyle\frac {f(x_1) - f(x_0)} {x_1 - x_0}$

If we let $x_1 = x_0 + h$ then the slope of the segment is

$\displaystyle\frac {f(x_0+h) - f(x_0)} {h}$

The Slope of $y=x^2$

Consider the task of computing the slope of the function $y=x^2$ at a given point.

Examples

Consider the function $y = f(x) = x^2$. In order to find the slope at a given point $(x_0 )$, we look at

$y = \displaystyle\frac{f (x_0 +h) - f(x_0)} {h}$

for small values of $h$.

For this particular function, $f (x) = x^2$, and hence

$f (x_0 +h) = (x_0 +h) ^2 = x^2 + 2hx_0 + h^2$

The slope of a line segment is therefore given by

$\displaystyle\frac{f (x_0 +h) - f(x_0)} {h}= \displaystyle\frac{2hx_0 + h^2} {h} = 2x_0 + h$

As we make $h$ steadily smaller, the segment slope, $2x_0 + h$, tends towards $2x_0$. It follows that the slope, $y'$, of the curve at a general point $x$ is given by $y' = 2x$.

The Tangent to a Curve

A tangent to a curve is a line that intersects the curve at exactly one point. The slope of a tangent for the function $y=f(x)$ at the point $(x_0,f(x_0))$ is

$\lim_{h\to0}\displaystyle\frac{f(x_0+h)-f(x_0)}{h}$

Details

To find the slope of the tangent to a curve at a point, we look at the slope of a line segment between the points $(x_0,f(x_0))$ and $(x_0+h,f(x_0+h))$, which is

$\displaystyle\frac{f(x_0+h)-f(x_0)}{h}$

and then we take $h$ to be closer and closer to $0$. Thus the slope is

$\lim_{h\to0}\displaystyle\frac{f(x_0+h)-f(x_0)}{h}$

when this limit exists.

Examples

Example

We wish to find the tangent line for the function $f(x)=x^2$ at the point $(1,1)$. First we need to find the slope of this tangent, it is given as

$\lim_{h\to0}\displaystyle\frac{(1+h)^2-1^2}{h}=\lim_{h\to0}\displaystyle\frac{2h+h^2}{h}=\lim_{h\to0}(2+h)=2$

Then, since we know the tangent goes through the point $(1,1)$ the line is $y=2x-1$.

The Slope of a General Curve

Details

Imagine a nonlinear function whose graph is a curve described by the equation $y = f(x)$. Here we want to find the slope of a line tangent to the curve at a specific point $(x_0)$. The slope of the line segment is given by the equation $\displaystyle\frac{f (x_0 +h) - f(x_0)} {h}$. Reducing $h$ towards zero, gives the slope of this curve if it exists.