Studdy AI Logo
Math  /  Algebra

Question4. Determine whether the following functions are symmetric to the xx-axis, yy-axis, or the origin. a. y=x5−6xy=x^{5}-6 x b. y2=x+4y^{2}=x+4

Studdy Solution

STEP 1

What is this asking? We need to figure out if these equations have any kind of symmetry, like mirroring across the x-axis, y-axis, or flipping around the origin. Watch out! Don't mix up the tests for different kinds of symmetry!
A graph can have more than one kind of symmetry, or even *no* symmetry at all.

STEP 2

1. Test for x-axis symmetry
2. Test for y-axis symmetry
3. Test for origin symmetry

STEP 3

To test for x-axis symmetry, we **replace** yy with −y-y in the original equation.
If we get an equivalent equation, then the graph is symmetric about the x-axis.

STEP 4

**Original equation:** y=x5−6xy = x^5 - 6x **Substitute** −y-y for yy: −y=x5−6x-y = x^5 - 6x **Multiply** both sides by −1-1 to isolate yy: y=−x5+6xy = -x^5 + 6x This is *not* the same as the original equation, so y=x5−6xy = x^5 - 6x is **not symmetric** about the x-axis.

STEP 5

**Original equation:** y2=x+4y^2 = x + 4 **Substitute** −y-y for yy: (−y)2=x+4(-y)^2 = x + 4 **Simplify:** y2=x+4y^2 = x + 4 This *is* the same as the original equation!
So, y2=x+4y^2 = x + 4 *is* **symmetric** about the x-axis.

STEP 6

Now, let's check for y-axis symmetry.
We **replace** xx with −x-x in the original equation.
If we get an equivalent equation, then the graph is symmetric about the y-axis.

STEP 7

**Original equation:** y=x5−6xy = x^5 - 6x **Substitute** −x-x for xx: y=(−x)5−6(−x)y = (-x)^5 - 6(-x) **Simplify:** y=−x5+6xy = -x^5 + 6x This is *not* the same as the original equation, so y=x5−6xy = x^5 - 6x is **not symmetric** about the y-axis.

STEP 8

**Original equation:** y2=x+4y^2 = x + 4 **Substitute** −x-x for xx: y2=−x+4y^2 = -x + 4 This is *not* the same as the original equation, so y2=x+4y^2 = x + 4 is **not symmetric** about the y-axis.

STEP 9

Finally, we test for symmetry about the origin.
We **replace** *both* xx with −x-x *and* yy with −y-y in the original equation.
If we get an equivalent equation, then the graph is symmetric about the origin.

STEP 10

**Original equation:** y=x5−6xy = x^5 - 6x **Substitute** −x-x for xx and −y-y for yy: −y=(−x)5−6(−x)-y = (-x)^5 - 6(-x) **Simplify:** −y=−x5+6x-y = -x^5 + 6x **Multiply** both sides by −1-1: y=x5−6xy = x^5 - 6x This *is* the same as the original equation!
So, y=x5−6xy = x^5 - 6x *is* **symmetric** about the origin.

STEP 11

**Original equation:** y2=x+4y^2 = x + 4 **Substitute** −x-x for xx and −y-y for yy: (−y)2=−x+4(-y)^2 = -x + 4 **Simplify:** y2=−x+4y^2 = -x + 4 This is *not* the same as the original equation, so y2=x+4y^2 = x + 4 is **not symmetric** about the origin.

STEP 12

y=x5−6xy = x^5 - 6x is symmetric about the **origin**. y2=x+4y^2 = x + 4 is symmetric about the **x-axis**.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord