Here are the step-by-step solutions for each polynomial function on your worksheet.
Problem 1: $f(x) = x^3 + 3x^2 - 4$
a) End Behavior
Look at the term with the highest power (the leading term), which is $x^3$.
* The degree is
3 (odd).
* The coefficient is
positive ($+1$).
*
Rule: For odd degrees with positive coefficients, the graph starts down on the left and goes up on the right.
*
Answer: As $x \to -\infty$, $f(x) \to -\infty$; as $x \to \infty$, $f(x) \to \infty$.
b) Real Zeros
Set $f(x) = 0$: $x^3 + 3x^2 - 4 = 0$.
We can test simple integer values like $\pm 1, \pm 2$.
* Try $x = 1$: $(1)^3 + 3(1)^2 - 4 = 1 + 3 - 4 = 0$. So,
$x = 1$ is a zero.
Since $x=1$ is a zero, we can divide the polynomial by $(x-1)$ to find the other factors.
$(x^3 + 3x^2 - 4) \div (x-1) = x^2 + 4x + 4$.
Now factor $x^2 + 4x + 4$. This is a perfect square: $(x+2)(x+2) = (x+2)^2$.
So the zeros come from $(x-1)(x+2)^2 = 0$.
*
Answer: The real zeros are
$1$ and $-2$.
c) Multiplicity and Turning Points
*
Multiplicity: Look at the exponents of the factors found in part (b).
* Zero at $x = 1$: Factor is $(x-1)^1$. Multiplicity is
1 (odd, so it crosses the axis).
* Zero at $x = -2$: Factor is $(x+2)^2$. Multiplicity is
2 (even, so it touches/bounces off the axis).
*
Turning Points: The maximum number of turning points is the degree minus 1. Degree is 3, so max turning points is $3 - 1 = 2$.
*
Answer: Multiplicity of $1$ is $1$; multiplicity of $-2$ is $2$. There are
2 turning points.
d) Intercepts and Graph Sketch
*
x-intercepts: These are the zeros:
$(1, 0)$ and $(-2, 0)$.
*
y-intercept: Set $x = 0$. $f(0) = 0^3 + 3(0)^2 - 4 = -4$. Point:
$(0, -4)$.
*
Sketch: Start from bottom left, go up through $(-2,0)$ but bounce back down (because of multiplicity 2), turn around at the y-intercept area, cross through $(1,0)$, and go up to the top right.
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Problem 2: $f(x) = -x^4 + 4x^3 - 4x^2$
a) End Behavior
Leading term is $-x^4$.
* The degree is
4 (even).
* The coefficient is
negative ($-1$).
*
Rule: Even degree with negative coefficient means both ends point down.
*
Answer: As $x \to -\infty$, $f(x) \to -\infty$; as $x \to \infty$, $f(x) \to -\infty$.
b) Real Zeros
Set $f(x) = 0$: $-x^4 + 4x^3 - 4x^2 = 0$.
Factor out the greatest common factor, which is $-x^2$:
$-x^2(x^2 - 4x + 4) = 0$.
Factor the inside part $(x^2 - 4x + 4)$ into $(x-2)(x-2)$ or $(x-2)^2$.
Equation becomes: $-x^2(x-2)^2 = 0$.
*
Answer: The real zeros are
$0$ and $2$.
**c) Multiplicity and Turning
Parent Tip: Review the logic above to help your child master the concept of polynomial graphs worksheet.
