**Key Terms**

- Cubic Function – A polynomial function of degree 3.
- The general form of a cubic function is , where a, b, c, and d are real numbers and a is not equal to zero.

- End Behavior – The way in which a graph of a function appears on the left and right sides of the Cartesian coordinate plane.
- It is the overall appearance of the graph.

- Estimate – To approximate or make a rough calculation.
- Extreme Value – A value of the function y = f(x) that is the greatest or least value of y.
- The greatest value is called the maximum, and the least value is called the minimum.

- Polynomial Function – A function that can be defined by evaluating an algebraic expression with one or more terms.
- None of the variables is in the denominator of a fraction, and any exponents are whole numbers.

- Roots – Values for which a function equals zero.
- The roots are also called zeros of the function.
- Any x-value at which the graph of a function crosses the x-axis is a root of the function.

**Review**

- Volume = length • width • height

- Graphs
- Read from left to right
- To graph a polynomial equation, plot some points and then connect them with a smooth curve.
- When you multiply
*all*the coefficients on the right side of an equation by -1, the graph gets flipped over the*x*-axis.

**Notes**

- Evaluating Functions

- Polynomial Graph Shapes
- Quadratics (highest degree: 2) and Quartics (highest degree: 4): parabola shapes
- Quadratics have a hill (when negative: flipped) or valley (when positive) in the graph
- Ex.

- Quartics also have a hill (when negative: flipped) or valley (when positive) in the graph, but they are wider with sharper curves
- Ex.
- Quartics can also have multiple hills and valleys
- Ex.

- Quadratics have a hill (when negative: flipped) or valley (when positive) in the graph
- Cubics (highest degree: 3) and Quintics (highest degree: 5): S-shapes
- Cubics are simple s-curves without hills or valleys
- Ex.

- Quintics are also s-curves, but are wider with sharper curves
- Ex.

- Both Cubics and Quintics can have multiple hills and valleys
- Ex.

- Cubics are simple s-curves without hills or valleys

- Quadratics (highest degree: 2) and Quartics (highest degree: 4): parabola shapes

- Polynomial Graph Properties
- Polynomials that are “not in their simplest form” have parent functions have been shifted, stretched, compressed, or flipped.
- The place where a polynomial’s graph crosses the y-axis (its y-intercept) is the equation’s constant.
- All polynomials have smooth, continuous graphs that do not cross over themselves.
- The plotted points must be connected by only one curving line without breaks.
- There is only one way to connect the graph’s plotted points.

- Degree – the highest exponent value of a polynomial
- Even degree: start high, end high (up)
- Odd degree: start low, end high (up)
- Positive leading coefficient: end high (up)
- Negative leading coefficient: end low (down)

- The bumps (hills and valleys) in the graph are based on having additional x-variables in the polynomial
- Ex. is a simple parabola with a valley, but some look like this: , and have a hill and two valleys (see Fourth-Degree Function below).
- Ex. is a simple s-curve, but some look like this: $, and have a hill and a valley (see Third-Degree Function below).
- Ex. Some polynomials have many hills and valleys: (see Fifth-Degree Function below)

- Extreme Values
- A value for which the polynomial is bigger or smaller than any nearby values.
- A maximum looks like the top of a hill
- A minimum looks like the bottom of a valley
- An absolute maximum is the highest point on the graph or in a given domain
- An absolute minimum is the lowest point on the graph or in a given domain
- A relative maximum is a hill in the graph, but it’s not the highest hill
- A relative minimum is a valley in the graph, but it’s not the lowest valley

- Maximum and Minimum Values
- The maximum of a polynomial is its greatest value and the highest point on its graph.
- You can estimate the extreme value or values by plotting points.
- First, plot enough points so you can see the shape of the graph.
- Then look for the area where you’d expect to find the maximum value and try to figure out what range of x-values the points in that area would have.
- The x-value that gives the greatest y-value is the best estimate for the graph’s maximum.

- Roots of a Polynomial
- Where the graph crosses the x-axis
- The x-values for which y = 0
- Also called zeros of the function
- For higher-order polynomials, it is possible to have as many roots as the degree of the function.

- Graphing to Estimate Roots
- Step 1: Use the scale and graph to identify a range for each root.
- Step 2: Try different x-values within that range.
- Step 3: When you find the x-value that gives y = 0, you have found the root.

**Examples**

- Ex 1. Evaluate the function below for x = 4.
- Substitute 4 for all x values:
- Simplify:
- Answer:

- Ex 2. At what values of x, does f(x) = 0?
- This is where all y-values equal 0 (the roots)

Answers: -1, 2, and 4