Question

f(x)=x^4 - 9x^2
a) use the leading coefficient test to determine the graphs end behavior.
a. the graph of f(x) rises left and rises right.
b. the graph of f(x) falls left and falls right.
c. the graph of f(x) falls left and rises right.
d. the graph of f(x) rises left and falls right.
b) find the x - intercepts.
x = - 3,0,3 (type an integer or a decimal. use a comma to separate answers as needed.)
at which zeros does the graph of the function cross the x - axis? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. x = - 3,3 (type an integer or a decimal. use a comma to separate answers as needed.)
b. there are no x - intercepts at which the graph crosses the x - axis.
at which zeros does the graph of the function touch the x - axis and turn around? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. x = 0 (type an integer or a decimal. use a comma to separate answers as needed.)
b. there are no x - intercepts at which the graph touches the x - axis and turns around.
c) find the y - intercept by computing f(0).
f(0)=0
d) determine the symmetry of the graph.
odd; origin symmetry
even; y - axis symmetry
neither

Explanation:

Step1: Analyze leading - coefficient and degree

The function $f(x)=x^{4}-9x^{2}$ is a polynomial of degree $n = 4$ (even) and leading - coefficient $a = 1$ (positive). According to the leading - coefficient test, for a polynomial $y = a_nx^n+\cdots+a_0$ with $n$ even and $a_n>0$, as $x\to-\infty$, $y\to+\infty$ and as $x\to+\infty$, $y\to+\infty$. So the graph of $f(x)$ rises left and rises right.

Step2: Find x - intercepts

Set $f(x)=0$, so $x^{4}-9x^{2}=x^{2}(x^{2} - 9)=x^{2}(x - 3)(x + 3)=0$. Solving $x^{2}=0$ gives $x = 0$, solving $x-3=0$ gives $x = 3$, and solving $x + 3=0$ gives $x=-3$.

Step3: Determine crossing/touching x - axis

For a factor of the form $(x - c)^k$ in the factored form of a polynomial: if $k$ is odd, the graph crosses the $x$-axis at $x = c$; if $k$ is even, the graph touches the $x$-axis at $x = c$. For $x^{2}(x - 3)(x + 3)$, the factor $(x - 3)$ and $(x + 3)$ have $k = 1$ (odd), so the graph crosses the $x$-axis at $x=-3,3$. The factor $x^{2}$ has $k = 2$ (even), so the graph touches the $x$-axis at $x = 0$.

Step4: Find y - intercept

Substitute $x = 0$ into $f(x)$. Then $f(0)=0^{4}-9\times0^{2}=0$.

Step5: Check for symmetry

A function $y = f(x)$ is even if $f(-x)=f(x)$ and odd if $f(-x)=-f(x)$. Calculate $f(-x)=(-x)^{4}-9(-x)^{2}=x^{4}-9x^{2}=f(x)$. So the function is even and has $y$-axis symmetry.

Answer:

a) A. The graph of f(x) rises left and rises right.
b) $x=-3,0,3$; $x=-3,3$; $x = 0$
c) $f(0)=0$
d) Even; y - axis symmetry