Unit 4: Polynomial Functions and Their Graphs
Reading a polynomial's degree and leading coefficient like a weather forecast for what its graph does at the far edges.
6 topics · 23 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
End Behavior and the Leading Term
- How it worksHow do the degree and sign of the leading term set the graph's direction on both far left and far right?
- Worked exampleFind the end behavior of f(x) = -2x5 + 3x2 - 1 on both sides of the graph.
- Common mistakeWhy doesn't a polynomial's end behavior depend at all on its lower-degree terms?
- Going furtherWhat changes about the end behavior when a polynomial's degree flips from odd to even?
Your turn1 of 2
For f(x) = -7x5 + 6x2 - 8, what happens to f(x) as x approaches positive infinity and as x approaches negative infinity?x-Intercepts and the Shape of a Polynomial Graph
- How it worksHow do you find a polynomial's x-intercepts by factoring and reading off each root?
- Worked exampleFind every x-intercept of f(x) = x3 - x2 - 6x.
- Common mistakeWhy doesn't a polynomial's y-intercept come from the same process as its x-intercepts?
- Going furtherWhat changes about finding x-intercepts when a polynomial is given in expanded form instead of already factored?
Your turn1 of 2
Find the sum of the two NONZERO x-intercepts of f(x) = x(x - 7)(x + 7).Turning Points and Local Extrema
- How it worksHow does a polynomial's degree cap the number of turning points its graph can have?
- Worked exampleFind how many turning points the graph of f(x) = x4 - 4x2 can have at most, and sketch it.
- Common mistakeWhy doesn't a polynomial of degree five have to actually use all four of its possible turning points?
- Ties togetherHow does the maximum number of turning points on a polynomial's graph connect to the maximum number of real zeros it can have?
Your turn1 of 2
What is the maximum possible number of turning points for a polynomial of degree 8?Graphing a Polynomial from Its Factors
- How it worksWalk me through combining end behavior, zeros, and multiplicity to sketch a polynomial's graph.
- Worked exampleSketch the graph of f(x) = (x + 2)(x - 1)2(x - 3) using its zeros and end behavior.
- Ties togetherHow does sketching a polynomial from its zeros and end behavior set up the same strategy used later to graph a rational function?
Your turn1 of 2
For f(x) = (x + 3)(x - 7)2(x - 12), which x-intercept does the graph touch and bounce at, rather than cross through?Even and Odd Polynomial Functions
- How it worksHow does checking f(-x) against f(x) tell you whether a polynomial is even, odd, or neither?
- Worked exampleFind whether f(x) = x4 - 3x2 + 5 is even, odd, or neither.
- Common mistakeWhy doesn't a polynomial need to have only even exponents to end up an even function?
- Going furtherWhat happens when you add an even function and an odd function together?
Your turn1 of 2
Is f(x) = 7x4 - 6x2 + 8 even, odd, or neither?Modeling with Polynomial Functions
- How it worksHow does cutting equal squares from the corners of a sheet and folding up the sides build a cubic volume formula?
- Worked exampleFind the volume function for a box folded from a 20 by 30 inch sheet with squares of side x cut from each corner.
- Going furtherWhat happens to the box's volume function if the sheet starts as a square instead of a rectangle?
- Ties togetherHow does restricting the domain of this volume model connect to excluding values from a rational function's domain?
Your turn1 of 2
A rectangular sheet measures 17 by 16 inches. Squares of side x are cut from each corner and the sides folded up. Write the volume V(x) of the resulting box as a function of x.