Graphing polynomials To graph a polynomial we need to know three things 1) Type of polynomial 2) Roots 3) y-intercept

Types of Polynomials Positive odd Negative odd Positive even Starts down ends up Negative odd Starts up ends down Positive even Up on both ends Negative even Down on both ends

We can find the type from the sign and power of the leading term. The sign tells us if it is positive or negative and the power (degree) tells us if it is even or odd. The degree is even Examples: The coefficient is positive Type: positive even

State the type for each. Negative even Negative odd Positive odd Positive even Negative even

Roots The degree of the polynomial, tells us how many roots we will have. Example: How many roots will the following functions have? 4 roots 5 roots 4 roots 6 roots

Roots Multiplicity of roots Single root Example: Factor (x+2) root: -2 graph goes straight through root double root Example: Factor (x-1)2 root: 1 (M2) graph goes through the root like a quadratic Indicates a double root triple root Example: Factor (x+4)3 root: -4 (M3) graph goes through the root like a cubic Indicates a triple root

Roots Roots: -5, 3, -2 Roots: 0, 4 (M2), Roots: 0 (M2), 3, -4 (M3) Real roots are x-intercepts. To find the roots, we let y = 0 and solve for x. Example: Find the roots for the following. Roots: -5, 3, -2 Roots: 0, 4 (M2), Roots: 0 (M2), 3, -4 (M3)

Y-intercept y-intercept: (0, -60) y-intercept: (0, 20) To find the y-intercept let x = 0 Always write the y-intercept as a point. Example: Find the y-intercept for each y-intercept: (0, -60) y-intercept: (0, 20) y-intercept: (0,0)

First, plot the roots and label Now we are ready to graph. State the type, roots, y-intercept and graph. First, plot the roots and label positive odd Type: ______________________ 3, -4, 1 roots: ______________________ (0, 24) y-intercept: __________ Last, sketch the graph (remember the type helps us with the shape) Next, plot and label the y-intercept

State the type, roots, y-intercept and graph. -3 is a double root, so it the graph looks like a quadratic here Negative even Type: ______________________ 0, -3 (M2), roots: ______________________ (0, 0) y-intercept: __________

State the type, roots, y-intercept and graph. -3 is a triple root, so it the graph looks like a cubic here positive odd Type: ______________________ 2, -4 (M3), -1 roots: ______________________ y-intercept: __________

State the type, roots, y-intercept and graph.

State the type, roots, y-intercept and graph.

State the type, roots, y-intercept and graph.

Graphing Polynomials Day 2

State the type, roots, y-intercept and graph. Expanded Form Partially Factored Sometimes we need to factor in order to find the roots. Factored form: ______________________________________________ Positive odd Type: ______________________ -2 (M2), 1 (M3) roots: ______________________ (0, -2) y-intercept: __________ Note: the type and y-intercept are both easy to find from the expanded form.

State the type, roots, y-intercept and graph. Remember to factor first Type: ______________________ roots: ______________________ y-intercept: __________

State the type, roots, y-intercept and graph.

Remember real roots are x-intercepts, but imaginary roots are not. Imaginary roots always come in pairs. Sketch a graph for each description. A negative even function with 3 real roots. A positive even function with no real roots A negative odd function with 5 roots.