Question

1. which graph shows the proper end behavior for the equation $y = x^4 + 35x^3 - 23x^2 - 43$ as $x$ becomes a large positive or negative number?

a.

b.

c.

d.

1. which of the following graph(s) does not show symmetry with the x-axis?

a.

b.

c.

d.

Explanation:

41 Step1: Identify leading term

Leading term is $x^4$ (even degree, positive coefficient).

41 Step2: Determine end behavior

As $x\to\pm\infty$, $x^4\to+\infty$, so both ends rise.

42 Step1: X-axis symmetry rule

A graph has x-axis symmetry if replacing $(x,y)$ with $(x,-y)$ leaves it unchanged.

42 Step2: Check each option

• A: $(x,y)\to(x,-y)$ changes the graph (not symmetric).
• B: $(x,y)\to(x,-y)$ changes the graph (not symmetric).
• C: $(x,y)\to(x,-y)$ changes the graph (not symmetric).
• D: $(x,y)\to(x,-y)$ leaves the circle unchanged (symmetric).

Thus, A, B, C do not show x-axis symmetry.

Answer:

1. A
2. A, B, C