Interactive Mathematics Graphics

Home Teaching Research Fun Math

Function Notation

Directions: Try moving around the top slider, which adjusts the input \(x\) which we are using to compute \(f(x)\).

Click to Show or Hide the Graph

Function Transformations

Click to Show or Hide the Graph of \(f(x) = a\sqrt{x+b}+c\)

Click to Show or Hide the Graph of \(f(x) = a(x+b)^2+c\)

Click to Show or Hide the Graph of \(f(x) = \dfrac{a}{x+b} + c\)

Click to Show or Hide the Graph of \(f(x) = a\cdot\sin\left(b\cdot x+c\right)+d\)

Linear Approximations and the Tangent Line

The line tangent to \(f(x)\) at \(a\) gives a good approximation to \(f(x)\) for \(x\) near \(a\). This interactive graphic can help you get a sense of how accurate this approximation is.

Directions: Try moving around the top slider, which adjusts the point \(x\) where we want to approximate \(f\). What happens to the error and percent error?
Now use the bottom slider to adjust the location \(a\) of the tangent line approximation to \(f\). For example, center the linear approximation to \(f\) at the point \(a= 0\). Now how does adjusting the point \(x\) impact the error and percent error?

Click to Show or Hide the Graph

The graphical meaning of \(f'(a)\)

There are two sides to the definition of \(f'(x)\). Graphically, \(f'(a)\) is the slope of the tangent to \(f(x)\) at \(a\). Algebraically, \(\displaystyle f'(a) = \lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h}\).

Directions: Move the slider to view the tangent line and \(f'(a)\) for different values of \(a\). What do you notice?

Click to Show or Hide the Graph

Using Secant Lines to Approximate the Tangent

Computing the slope of the line tangent to \(f(x)\) at \(a\) is tricky. Our formula for the slope of a line requires two points, but the tangent is defined using a single point.

As \(h\) gets smaller, the slope of the secant line between \((a,f(a))\) and \((a+h,f(a+h))\) approaches the slope of the tangent to \(f(x)\) at \(a\).
When \(h\) is very small, the slope of the secant is nearly identical to the slope of the tangent.

Directions: Try moving the slider for \(h\). What happens when you make \(h\) closer to 0? Next try moving the slider for \(a\), to approximate the tangent at a different point.

Click to Show or Hide the Graph

Compound Interest

Understanding the impact of different interest rates is an important life skill. Use this graphic to get a concrete picture of the impact of the interest on the principal.

Directions: The top slider controls the interest rate (as a decimal), the middle slider controls the number of compoundings per year, and the bottom slider controls the principle (in thousands).


Click to Show or Hide the Graph

General Exponential Functions

Changing the base of the exponential changes the graph significantly.

Directions: Move the slider around to examine the graph of \(y=a^x\) for different values of \(a\). What happens when \(a > 1\)? What about when \(a = 1\), or when \(a < 1\)?

Click to Show or Hide the Graph