6.2 Evaluate and Graph Polynomial Functions How do you identify and evaluate polynomial functions? What is synthetic substitution? How do you graph polynomial functions?

Polynomial Function f(x) = anxn + an−1xn−1+ … + a1x + a0 an ≠ 0 Exponents are all whole numbers Coefficients are all real numbers an is the leading coefficient a0 is the constant term n is the degree f(x) = 7x3 + 9x2 + 3x + 1 f(x) = a3x3 + a3−1x3−1 + a3−2x3−2 +1

Polynomial Functions Degree Type Standard form 0 Constant f(x) = a0 1 Linear f(x) = a1x + a0 2 Quadratic f(x) = a2x2 + a1x + a0 3 Cubic f(x) = a3x3 + a2x2 + a1x + a0 4 Quartic f(x) = a4x4 + a3x3 + a2x2 + a1x + a0

Identify polynomial functions Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. a. h (x) = x4 – x2 + 3 1 4 SOLUTION a. The function is a polynomial function that is already written in standard form. It has degree 4 (quartic) and a leading coefficient of 1.

Is the function a polynomial function Is the function a polynomial function? If so, write in standard form, state the degree, type and leading coefficient. f(x) = ½x2 − 3x4 −7 f(x) = x3 +3x f(x) = 6x2 + 2x−1 + x f(x) = −0.5x + πx2 −√2  

A polynomial function such as f(x) = 5x4 − 4x3 + 3x2 +2x + 1 has the following properties: A coefficient is the number in front of a variable. The leading coefficient is in front of the variable with highest power. Leading Coefficient: 5 Constant Term: 1 A Constant Term does not have a variable. The degree is the highest power of a polynomial. Degree: 4 The terms are written from the highest power to the lowest power. Standard Form: All powers are whole numbers. No decimals or fractions or negative powers are allowed in a polynomial power. Exponents:

f(x) = 2 f(x) = x + 1 Degree = 0 Degree = 1 No Zeros 1 Zero Constant y y x x Constant Function Linear Function Degree = 0 Degree = 1 No Zeros 1 Zero

f(x) = x2 + 2x − 3 f(x) = x3− 5x + 2 Degree = 2 Degree = 3 2 Zeros y y x x Quadratic Function Cubic Function Degree = 2 Degree = 3 2 Zeros 3 Zeros

f(x) = x4 − 3x3 + 2x2 +7x + 1 f(x) = x5− 5x3 + 4x Degree = 4 y y x x Quartic Function Quintic Function Degree = 4 Degree = 5 4 Zeros 5 Zeros

No, contains fractional variable. Determine if each expression is a polynomial function. If so, write it in standard form and state the degree. Yes, the degree is 4. Yes, the degree is 5. No, contains fractional variable. No, contains variable exponent.

Synthetic Substitution Use synthetic substitution to evaluate f(x) = 2x4 −8x2 + 5x −7 when x = 3

Direct Substitution vs Synthetic Substitution Direct substitution is replacing a variable with a number in a function to find the value of that function at that point. If x = 3 what is the value of y (or f(x)). Synthetic substitution is another way to evaluate a polynomial function that involves fewer operations than direct substitution.

Direct Substitution Example Use direct substitution to evaluate f (x) = 2x4 – 5x3 – 4x + 8 when x = 3. f (x) = 2x4 – 5x3 – 4x + 8 Write original function. f (3) = 2(3)4 – 5(3)3 – 4(3) + 8 Substitute 3 for x. = 162 – 135 – 12 + 8 = 23 Simplify

Try this one… 5. g (x) = x 3 – 5x2 + 6x + 1; x = 4 SOLUTION Write original function. g (x) = 4 3 – 5(4)2 + 6(4) + 1 Substitute 4 for x. = 64 – 80 + 24 + 1 Evaluate and multiply. = 9 Simplify Be careful when you put this into your calculator, especially when you change between adding/subtracting and multiplying. You may find it is easier to put parts in and record the values, then add/subtract in the last step.

Use direct substitution to evaluate the polynomial functions for the value of x.

Synthetic Substitution Use synthetic substitution to evaluate f (x) from Example 2 when x = 3. f (x) = 2x4 – 5x3 – 4x + 8 SOLUTION STEP 1 Write the coefficients of f (x) in order of descending exponents. Write the value at which f (x) is being evaluated to the left.

STEP 2 Bring down the leading coefficient. Multiply the leading coefficient by the x-value. Write the product under the second coefficient. Add. STEP 3 Multiply the previous sum by the x-value. Write the product under the third coefficient. Add. Repeat for all of the remaining coefficients. The final sum is the value of f(x) at the given x-value.

Synthetic substitution gives f(3) = 23, which matches the result shown previously ANSWER

Try this one… Use synthetic substitution to evaluate the polynomial function for the given value of x. 6. f (x) = 5x3 + 3x2 – x + 7; x = 2 STEP 1 Write the coefficients of f (x) in order of descending exponents. Write the value at which f (x) is being evaluated to the left.

Synthetic substitution gives f(2) = 57 ANSWER STEP 2 Bring down the leading coefficient. Multiply the leading coefficient by the x-value. Write the product under the second coefficient. Add. STEP 3 Multiply the previous sum by the x-value. Write the product under the third coefficient. Add. Repeat for all of the remaining coefficients. The final sum is the value of f(x) at the given x-value. Synthetic substitution gives f(2) = 57 ANSWER

Try this one… – 1 – 2 – 1 0 4 – 5 – 1 – 2 – 1 0 4 – 5 2 –1 1 –5 7. g (x) = – 2x4 – x3 + 4x – 5; x = – 1 – 1 – 2 – 1 0 4 – 5 2 –1 1 –5 – 1 – 2 – 1 0 4 – 5 – 2 1 –1 5 – 10 Synthetic substitution gives f(– 1) = – 10 ANSWER

Use synthetic substitution to evaluate the polynomial functions for the value of x. 3 4 −10 −3 5 −2 + 12 6 9 42 × 4 2 3 14 40

Use synthetic substitution to evaluate the polynomial functions for the value of x. 2 −2 0 3 −5 7 −9 −4 −8 −10 −30 −46 −2 −4 −5 −15 −23 −55

Use synthetic substitution to evaluate the polynomial functions for the value of x. − 2 −5 6 −2 −7 2 −3 10 −32 68 −122 240 −5 16 −34 61 −120 237 p

Questions?

End Behavior The end behavior of a polynomial graph is what happens to the graph as it continues past the physical graph. There are arrows on the ends of the graph that shows the direction in which the graph continues. The graph approaches positive infinity (+∞) or negative infinity (−∞). x→+∞ is read as “x approaches positive infinity.

The degree is odd and the leading coefficient is positive The degree is odd and the leading coefficient is positive. So, f (x) → – ∞ as x → – ∞ and f (x) → + ∞ as x → + ∞. The degree is odd and leading coefficient is negative. So, f (x) → + ∞ as x → – ∞ and f (x) → – ∞ as x → + ∞. The degree is even and leading coefficient is positive. So, f (x) → ∞ as x → – ∞ and f (x) → ∞ as x → + ∞. The degree is even and the leading coefficient is negative. So, f (x) → – ∞ as x → – ∞ and f (x) → – ∞ as x → + ∞.

How will I EVER remember this?!? The degree of a line is 1 which is odd, the slope is negative. The degree of a line is 1 which is odd, the slope is positive. CV CV The degree of a parabola is 2 which is even, “a” is positive. The degree of a parabola is 2 which is even, “a” is negative.

Try this one… 8. Describe the degree and leading coefficient of the polynomial function whose graph is shown. ANSWER degree: odd, leading coefficient: negative

From the graph, f (x) → – ∞ as x → – ∞ and f (x) → – ∞ as x → + ∞ From the graph, f (x) → – ∞ as x → – ∞ and f (x) → – ∞ as x → + ∞. So, the degree is even and the leading coefficient is negative. The correct answer is D. ANSWER

End Behavior of a Polynomial Function’s Graph x y x y

End Behavior of a Polynomial Function’s Graph x y x y

Describe the end behavior of the polynomial function’s graph. x f(x) -4 -3 -2 -1 1 2 5

Graph x y x f(x) -4 -3 -2 -1 1 2 p

Questions?