5.2 – Polynomial Functions

Key Terms

• Cubic Function – A polynomial function of degree 3.
• The general form of a cubic function is $F(x)=ax^3+bx^2+cd+d$, 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. $y=x^2$
• Quartics also have a hill (when negative: flipped) or valley (when positive) in the graph, but they are wider with sharper curves
• Ex. $y=x^4$
• Quartics can also have multiple hills and valleys
• Ex. $y=x^4-2x^2$
• Cubics (highest degree: 3) and Quintics (highest degree: 5): S-shapes
• Cubics are simple s-curves without hills or valleys
• Ex. $y=x^3$
• Quintics are also s-curves, but are wider with sharper curves
• Ex. $y=x^5$
• Both Cubics and Quintics can have multiple hills and valleys
• Ex. $y=x^3-2x^2$

• 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. $y=x^2$ is a simple parabola with a valley, but some look like this: $y=x^4+2x^3$, and have a hill and two valleys (see Fourth-Degree Function below).
• Ex. $y=x^3$ is a simple s-curve, but some look like this: $y=x^3-2x$\$, and have a hill and a valley (see Third-Degree Function below).
• Ex. Some polynomials have many hills and valleys: $y=x^5-5x^4+5x^3+5x^2-5.5x-1$ (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.
• $F(x)=x^3+2x^2+1$
• Substitute 4 for all x values: $F(4)=4^3+2(4)^2+1$
• Simplify: $F(4)=64+32+1$
• Answer: $F(4)=97$

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