Find a cubic function y=a x^3+b x^2+c x+d whose graph has...

Key Concepts

Cubic Polynomial

A cubic polynomial is a third degree polynomial typically written in the form y = ax³ + bx² + cx + d. Its graph can have one or two turning points, and its overall shape is determined by its coefficients. This concept is central in understanding the behavior of polynomial functions and how their terms influence the graph's curvature and inflection.

Derivative

The derivative of a function represents its instantaneous rate of change and is used to determine the slope of the function at any point. For polynomial functions, the derivative is calculated using power rules and provides critical information on the locations of peaks, troughs, and points of inflection in the graph.

Horizontal Tangents

A horizontal tangent on a function's graph indicates a point where the slope is zero. This occurs when the derivative of the function is equal to zero. Recognizing and setting up this condition is key for identifying local extrema or points of inflection, which are important in the study of function behavior.

System of Equations

When a function must satisfy specific conditions—such as passing through given points and having horizontal tangents at those points—a system of equations is established. Each condition contributes an equation (from either the function itself or its derivative) so that the set of unknown coefficients can be solved simultaneously, showcasing the interconnected nature of function values and rates of change.

Transcript

00:01 We want to find a, b, c, and d such that the graph of y has horizontal tangents at the given points.

00:11 So horizontal tangents means that the line tangent to the curve is horizontal, which means that the slope of the tangent line is zero, okay, and the slope of the tangent line is just dy dx at that point.

00:31 So we need to find a, b, c, and d such that dy dx at the point negative 2, 6 is zero and dy dx at the point 2, 0 is zero.

00:52 Okay, we also need the graph to pass through these two points.

00:56 So we have four conditions total that this derivative is zero, this derivative is zero, and that it passes, the graph passes through this point and this point.

01:09 Okay, so those four conditions will give us four equations to solve for these four unknowns a, b, c, and d.

01:16 Okay, so let's start by calculating the derivative.

01:20 We have dy dx is equal to 3ax squared plus 2bx plus c.

01:32 So let's look at when d, when x is equal to negative 2.

01:40 Okay, then the derivative becomes 3a times negative 2 squared plus 2b times negative 2 plus c, which is equal to 12a minus 4b plus c.

01:55 Okay, and then we need this to equal zero.

01:58 So we have zero equals 12a minus 4b plus c.

02:04 Okay, now let's find the derivative when x is equal to 2.

02:10 Okay, so we have dy dx when x is 2 is equal to 3a times 2 squared plus 2b times 2 plus c, which is equal to 12a plus 4b plus c, and this is also equal to zero.

02:33 So here are two equations that we have so far.

02:39 I'm going to subtract them from each other, okay, to eliminate a and c.

02:45 So i get zero minus zero is zero, 12a minus 12a is zero, negative 4b minus 4b is negative 8b, and c minus c is zero.

02:56 Okay, so we're left with, this tells us that b equals zero.

03:01 Okay, so this is the first solution.

03:04 We need to find the values of the other three parameters.

03:09 So now let's plug in the points into our equation.

03:12 I'm going to do this up here, so i'll make some space.

03:17 We have the points there, so let's start with negative 2, 6.

03:24 So we have 6 equals a times negative 2 cubed.

03:28 Remember b is zero, so that term just goes away, okay, because this is zero.

03:35 And then we have plus c times negative 2 plus d.