Question 1

Which of the following statements correctly describes the symmetry of the graph of $$x = y^2$?

Accepted Answer

It is symmetric with respect to the x-axis.

Question 2

Given the function $$f(x) = x^2$ - 5$$, which type of symmetry does its graph exhibit?

Accepted Answer

Symmetry with respect to the y-axis.

Question 3

For the equation $$x^2$$ = $$y^2 + 2$, which algebraic test confirms its symmetry with respect to the origin?

Accepted Answer

Replace $$x$$ with $$-x$$ and $$y$$ with $$-y$$; the equation remains unchanged.

Question 4

Which of the following functions is even?

Accepted Answer

$$f(x) = x^2$

Question 5

A function $$f is $$called odd if for every $$x in $$its domain, which of the following holds?

Accepted Answer

$$f(-x) = -f(x)$$

Question 6

Determine whether the function $$f(x) = 5x^3$ - 6x^2$ - 2x is $$even, odd, or neither.

Accepted Answer

Odd

Question 7

Which algebraic test would you use to check if a graph is symmetric with respect to the y-axis?

Accepted Answer

Replace $$x$$ with $$-x$$ and check if the equation is unchanged.

Question 8

If a graph is symmetric with respect to the x-axis, what does this mean about the points $$(x, y)$$ and $$(x, -y)$$?

Accepted Answer

Both points are on the graph.

Question 9

Which of the following is true for the function $$f(x) = x^4$ - 2x^2$?

Accepted Answer

It is an even function.

Question 10

Suppose a chemical reaction rate is modeled by $$r(t) = t^3$ - t. Is$$ $$r(t) an $$even, odd, or neither function?

Accepted Answer

Odd

Question 11

A medical imaging device uses the function $$g(x) = x^2$ + 2 to $$model intensity. Which symmetry does its graph have?

Accepted Answer

Symmetry with respect to the y-axis.

Question 12

If $$f(x) is an $$even function, which of the following must be true for all $$x in $$its domain?

Accepted Answer

$$f(-x) = f(x)$$

Question 13

Which of the following equations is symmetric with respect to the origin?

Accepted Answer

$$y = x^3$

Question 14

A graph is rotated $$180^	ext{o}$$ about the origin and coincides with itself. What type of symmetry does it have?

Accepted Answer

Symmetry with respect to the origin.

Question 15

Which of the following is a necessary condition for a function to be both even and odd?

Accepted Answer

The function must be $$f(x) = 0$$ for all $$x.$$

Question 16

Given $$f(x) = x^2$ + 2$$, what is $$f(-x)$$ and what does this tell you about the function?

Accepted Answer

$$f(-x) = x^2$ + 2$$, so $$f is $$even.

Question 17

A real-world scenario: The height $$h(t) of a $$projectile is modeled by $$h(t) = -16t^2$ + 32t + 5. $$What symmetry does this function have?

Accepted Answer

No symmetry.

Question 18

Which of the following is the correct algebraic test for symmetry with respect to the origin?

Accepted Answer

Replace $$x$$ with $$-x$$ and $$y$$ with $$-y$$; if the equation is unchanged, the graph is symmetric with respect to the origin.

Question 19

If $$f(x) = x^5$ - x^3$, determine whether f$ is even, odd, or neither.

Accepted Answer

Odd

Question 20

A graph of $$y = x^2$ + 2 is $$folded along the y-axis. What happens?

Accepted Answer

The left and right sides coincide, showing y-axis symmetry.

Symmetry of Graphs and Functions: Section 2.4 Study Notes

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