What points help identify a cubic and an exponential graph?

Question

What are the different points on the graph we need to know which would help determine the equation of the curve. For example in a quadratic graph, we can determine the equation if we know either any three points, vertex and y intercept, two roots and a point, y intercept and two roots, etc. Similarly what are the various points on the graph which would help determine the graph for a cubic and exponential graph?

Answer 1

Let's discuss the case of a cubic. A cubic equation has the form $f(x)=ax^3 + bx^2+cd + d$. To fully determine the function, you need four pieces of information that will allow you to calculate $a,b,c,d$.

The simplest example of "four pieces of information" is the values of $f$ at four points, say $x_1$, $x_2$, $x_3$, $x_4$. If we know that $$ f(x_i) = y_i \quad \text{for} \; i=1,2,3,4 $$ then we have four linear equations that we can solve to get $a,b,c,d$. This is related to something called Lagrange interpolation.

An especially easy case is where three of the values are zeros of $f$. If we have $f(x_1) = f(x_2) = f(x_3)=0$, then we know that $f(x)=k(x-x_1)(x-x_2)(x-x_3)$ for some $k$. A fourth known value of $f$ will then allow you to calculate $k$.

Another common case is where we know the value and first derivative of $f$ at two points. This again gives us a system of four linear equations that we can solve to get $a,b,c,d$. This is related to so-called Hermite interpolation.

Another example: maybe you know the value of $f$ and its first, second, and third derivatives at some particular point. This lets you construct the Taylor series for $f$ at this point, and that Taylor series is actually identical to $f$ itself.

The four pieces of information can be a diverse mixture. For example, if you know that $f(0)=0$, $f'(0)=0$, $f(1) = 2$, and $f''(2) = 3$, then we have \begin{align} f(0) = 0 &\;\Longrightarrow\; d=0 \\ f'(0) = 0 &\;\Longrightarrow\; c=0 \\ f(1) = 2 &\;\Longrightarrow\; a+b+c+d =2 \\ f''(2) = 3 &\;\Longrightarrow\; 12a+4b = 3 \end{align} which, again, you can solve to get $a,b,c,d$.

So, in short, you need 4 pieces of information, which are typically values of the function or its derivative(s).