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MHF4U: Advanced Functions
Equations and Graphs of Polynomial Functions
J. Garvin
Slide 1/
Polynomial Functions In Factored Form
Polynomials are generally written in standard form, such as f (x) = x^3 + 4x^2 + x − 6. A more useful way to write a polynomial function’s equation is to use factored form, such as f (x) = (x − 1)(x + 2)(x + 3). Each factor corresponds to an x-intercept of the function. Factored Form of a Polynomial Function An equation of a polynomial function is in factored form if it is written as f (x) = a(x − r 1 )(x − r 2 )... (x − rn), where (x − rk ) is a factor corresponding to x-intercept rk.
Note that it is not always possible to express a polynomial function using factored form.
J. Garvin — Equations and Graphs of Polynomial FunctionsSlide 2/
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Polynomial Functions In Factored Form
Example Identify the factors, and x-intercepts, of the polynomial function f (x) = − 2 x(x + 5)(x − 3).
f (x) has three factors: x, x + 5 and x − 3.
These factors correspond to x-intercepts 0, −5 and 3.
J. Garvin — Equations and Graphs of Polynomial Functions Slide 3/
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Order of a Factor
Examine the x-intercepts of f (x) = 6x(x − 1)^2 (x − 2)^3 below.
How does the function behave around each x-intercept?
J. Garvin — Equations and Graphs of Polynomial Functions Slide 4/
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Order of a Factor
The function f (x) = 6x(x − 1)^2 (x − 2)^3 has x-intercepts at 0, 1 and 2.
At x = 0, the function changes from positive to negative, passing through the x-axis.
The function remains negative on either side of x = 1, “bouncing” off of the x-axis.
At x = 2, the function changes from negative to positive, passing through the x-axis.
How does this behaviour relate to the factors of the function?
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Order of a Factor
In two cases, x = 0 and x = 2, the exponents were odd. Both of these cases saw the function change from positive to negative, or vice versa. In the other case, x = 1, the exponent was even. No change in sign occurred here. Order of a Factor The factor (x − r )n^ has order n. If n is odd, the function crosses the x-axis at r. If n is even, the function touches (but does not cross) the x-axis at r.
Used in conjunction with a function’s end behaviour, identifying the order of each factor is a useful tool for sketching graphs.
Graphs of Polynomial Functions
Example
Sketch a graph of f (x) = (x + 3)^2 (x − 1).
f (x) has two distinct x-intercepts, at x = −3 and x = 1.
Another way to write the equation is f (x) = (x + 3)(x + 3)(x − 1). Multiplying all terms containing x, we obtain x^3 , so f (x) has degree 3 (cubic).
The leading coefficient is positive, so f (x) has Q3-Q1 end behaviour. Therefore, f (x) is negative as x → −∞.
Moving from left to right, the first x-intercept is at x = −3, where it has order 2. Thus, the function touches the x-axis at x = −3, but stays negative beyond it.
The next x-intercept is at x = 1, where it has order 1. f (x) changes from negative to positive at x = 1. J. Garvin — Equations and Graphs of Polynomial FunctionsSlide 7/
Order of a Factor
One final piece point is the y -intercept, which can be found by multiplying all of the constant terms and the leading coefficient. In this case, the y -intercept is (3)(3)(−1) = −9.
J. Garvin — Equations and Graphs of Polynomial FunctionsSlide 8/
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Graphs of Polynomial Functions
Example
Sketch a graph of f (x) = − 2 x(x + 1)(x − 2)^2.
f (x) has three distinct x-intercepts, at x = −1, x = 0 and x = 2.
f (x) is a quartic function, with degree 4.
The leading coefficient is negative, so f (x) has Q3-Q4 end behaviour. f (x) is negative as x → −∞.
From left to right, f (x) changes from negative to positive at x = −1, changes from positive to negative at x = 0, and touches the x-axis at x = 2.
The y -intercept is −2(0)(1)(−2)(−2) = 0.
J. Garvin — Equations and Graphs of Polynomial Functions Slide 9/
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Graphs of Polynomial Functions
J. Garvin — Equations and Graphs of Polynomial Functions Slide 10/
p o l y n o m i a l f u n c t i o n s
Graphs of Polynomial Functions
Example Given the graph of f (x) below, state the minimum possible degree, sign of the leading coefficient, factors, x-intercepts and intervals where the function is positive or negative.
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Graphs of Polynomial Functions
f (x) has Q2-Q4 end behaviour, so the degree must be odd and the leading coefficient is negative. There are x-intercepts at x = −3 (even order), x = 0 (odd order) and x = 2 (even order), so the minimum degree is 5. This is confirmed by the fact that there are 4 local minimums and maximums. f (x) is positive on the intervals (−∞, −3) ∪ (− 3 , 0). Since zero is neither positive nor negative, it is not included in the interval. f (x) is negative on the intervals (0, 2) ∪ (2, ∞).
