(a) Graph the function P(x)=(x-1)(x-3)(x-4) and find all...

(a) Graph the function $P(x)=(x-1)(x-3)(x-4)$ and find all local extrema, correct to the nearest tenth.
(b) Graph the function
$$Q(x)=(x-1)(x-3)(x-4)+5$$
and use your answers to part (a) to find all local extrema, correct to the nearest tenth.

Key Concepts

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication, and raised to non-negative integer exponents. Understanding their structure—including degree and leading coefficient—is essential for predicting their long?term behavior, number of turning points, and overall graph shape.

Local Extrema

Local extrema are points on a function’s graph where the function reaches a local minimum or maximum. They are identified by locating the critical points where the derivative is zero or undefined, and then determining which of these points correspond to local minima or maxima, often using second derivative tests or analysis of the derivative’s sign changes.

Derivatives and Critical Points

The derivative of a function provides information about the function’s rate of change. Setting the derivative equal to zero or identifying where it is undefined yields critical points, which are potential candidates for local extrema. This process is fundamental in calculus for optimizing functions and analyzing their behavior.

Function Transformations and Vertical Shifts

Function transformations involve changing the graph of a function in a systematic way, such as translating, stretching, or reflecting it. A vertical shift, achieved by adding a constant to the function, moves the entire graph up or down without altering the shape, thereby affecting the absolute values of local extrema while preserving their x-coordinates.

Transcript

00:03 All right.

00:04 We're talking about the graph of p of x, which is equal to x minus 1 times x minus 3 times x plus or x minus 4.

00:10 So we know we're going to have three roots.

00:14 Okay.

00:14 And this is going to be a third degree polynomial.

00:16 If i multiply x times x times x, the lead term or the lead coefficient would be 1x to the third power.

00:25 So it's the third degree polynomial.

00:27 It means my graph is decreasing to the left and it's increasing to the right.

00:32 I know that because the lead coefficient is positive.

00:36 Okay.

00:39 So this tells me that if i were taking a quick look at this sketch, then i'm going to have zeros at positive 1, positive 3, and positive 4.

00:55 I'm going to have a y intercept down here at negative 12.

00:59 And because i'm going down to the left and i'm going up to the right, then my curve has to be doing something, kind of.

01:10 Kind of like that.

01:11 Now, what i'm not 100 % sure about, i'm going to have two extreme.

01:16 I'm going to have a maximum, and i'm going to have a minimum here.

01:19 What i'm not 100 % sure about is where exactly those coordinates lie.

01:24 So i'm going to utilize the help of decimal graphing for this one.

01:29 And if i type this in real quick, so i do x minus 1 times x minus 3 times x minus 4.

01:47 And we see we get this curve here, and that we have a maximum at approximately 1 .8, 2 .1, and approximately 3 .6, negative .6...

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