Math / Geometry

QuestionUse possible symmetry to determine whether the graph is the graph of an even function, an odd function, or a function that is neither even nor odd.

Studdy Solution

STEP 1

What is this asking? Does this graph represent an even function, an odd function, or neither? Watch out! Don't mix up even and odd functions!
Symmetry across the y-axis means even, symmetry about the origin means odd.

STEP 2

1. Check for even symmetry.
2. Check for odd symmetry.

STEP 3

An even function has the property $f(-x) = f(x)$ .
This means if we fold the graph along the y-axis, the two sides should match up perfectly!

STEP 4

Let's look at the points on our graph.
We have the point $(-2, 1)$ .
If we change the sign of the x-coordinate, we get $(2, 1)$ , which is also on the graph!

STEP 5

We also see the points $(-1, 0)$ and $(1, 0)$ .
Changing the sign of $x$ in $(-1, 0)$ gives us $(1, 0)$ , which is on the graph!
Same with $(0, 1)$ .
Changing the sign of the x-coordinate gives us $(0, 1)$ , which is already on the graph!

STEP 6

It looks like for every point $(x, y)$ on the graph, the point $(-x, y)$ is also on the graph!
This means the graph is symmetric with respect to the y-axis.

STEP 7

An odd function has the property $f(-x) = -f(x)$ .
This means if we rotate the graph 180 degrees around the origin, it should look exactly the same!

STEP 8

Let's test a point.
We have the point $(1, 0)$ on our graph.
If we change the sign of both the x and y coordinates, we get $(-1, 0)$ .
This point is on the graph!

STEP 9

However, let's try another point.
We have $(2, 1)$ on the graph.
Changing the sign of both coordinates gives us $(-2, -1)$ .
This point is not on the graph!

STEP 10

Since the graph doesn't have symmetry about the origin, it's not an odd function.

STEP 11

The graph is symmetric with respect to the y-axis, but not the origin.
Therefore, the graph represents an even function.

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QuestionUse possible symmetry to determine whether the graph is the graph of an even function, an odd function, or a function that is neither even nor odd.

Studdy Solution

STEP 1

What is this asking? Does this graph represent an even function, an odd function, or neither? Watch out! Don't mix up even and odd functions!
Symmetry across the y-axis means even, symmetry about the origin means odd.

STEP 2

1. Check for even symmetry.
2. Check for odd symmetry.

STEP 3

An even function has the property $f(-x) = f(x)$ .
This means if we fold the graph along the y-axis, the two sides should match up perfectly!

STEP 4

Let's look at the points on our graph.
We have the point $(-2, 1)$ .
If we change the sign of the x-coordinate, we get $(2, 1)$ , which is also on the graph!

STEP 5

We also see the points $(-1, 0)$ and $(1, 0)$ .
Changing the sign of $x$ in $(-1, 0)$ gives us $(1, 0)$ , which is on the graph!
Same with $(0, 1)$ .
Changing the sign of the x-coordinate gives us $(0, 1)$ , which is already on the graph!

STEP 6

It looks like for every point $(x, y)$ on the graph, the point $(-x, y)$ is also on the graph!
This means the graph is symmetric with respect to the y-axis.

STEP 7

An odd function has the property $f(-x) = -f(x)$ .
This means if we rotate the graph 180 degrees around the origin, it should look exactly the same!

STEP 8

Let's test a point.
We have the point $(1, 0)$ on our graph.
If we change the sign of both the x and y coordinates, we get $(-1, 0)$ .
This point is on the graph!

STEP 9

However, let's try another point.
We have $(2, 1)$ on the graph.
Changing the sign of both coordinates gives us $(-2, -1)$ .
This point is not on the graph!

STEP 10

Since the graph doesn't have symmetry about the origin, it's not an odd function.

STEP 11

The graph is symmetric with respect to the y-axis, but not the origin.
Therefore, the graph represents an even function.

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QuestionUse possible symmetry to determine whether the graph is the graph of an even function, an odd function, or a function that is neither even nor odd.

Studdy Solution

STEP 1

What is this asking? Does this graph represent an even function, an odd function, or neither? Watch out! Don't mix up even and odd functions!Symmetry across the y-axis means even, symmetry about the origin means odd.

STEP 2

1. Check for even symmetry.2. Check for odd symmetry.

STEP 3

An even function has the property f(−x)=f(x)f(-x) = f(x)f(−x)=f(x).This means if we fold the graph along the **y-axis**, the two sides should match up perfectly!

STEP 4

Let's look at the points on our graph.We have the point (−2,1)(-2, 1)(−2,1).If we change the sign of the x-coordinate, we get (2,1)(2, 1)(2,1), which is also on the graph!

STEP 5

We also see the points (−1,0)(-1, 0)(−1,0) and (1,0)(1, 0)(1,0).Changing the sign of xxx in (−1,0)(-1, 0)(−1,0) gives us (1,0)(1, 0)(1,0), which is on the graph!Same with (0,1)(0, 1)(0,1).Changing the sign of the x-coordinate gives us (0,1)(0, 1)(0,1), which is already on the graph!

STEP 6

It looks like for every point (x,y)(x, y)(x,y) on the graph, the point (−x,y)(-x, y)(−x,y) is *also* on the graph!This means the graph is symmetric with respect to the **y-axis**.

STEP 7

An odd function has the property f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x).This means if we rotate the graph **180 degrees** around the **origin**, it should look exactly the same!

STEP 8

Let's test a point.We have the point (1,0)(1, 0)(1,0) on our graph.If we change the sign of *both* the x and y coordinates, we get (−1,0)(-1, 0)(−1,0).This point *is* on the graph!

STEP 9

However, let's try another point.We have (2,1)(2, 1)(2,1) on the graph.Changing the sign of both coordinates gives us (−2,−1)(-2, -1)(−2,−1).This point is *not* on the graph!

STEP 10

Since the graph doesn't have symmetry about the origin, it's *not* an odd function.

STEP 11

The graph is symmetric with respect to the y-axis, but not the origin.Therefore, the graph represents an **even** function.

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What is this asking? Does this graph represent an even function, an odd function, or neither? Watch out! Don't mix up even and odd functions!
Symmetry across the y-axis means even, symmetry about the origin means odd.

1. Check for even symmetry.
2. Check for odd symmetry.

An even function has the property $f(-x) = f(x)$ .
This means if we fold the graph along the y-axis, the two sides should match up perfectly!

Let's look at the points on our graph.
We have the point $(-2, 1)$ .
If we change the sign of the x-coordinate, we get $(2, 1)$ , which is also on the graph!

We also see the points $(-1, 0)$ and $(1, 0)$ .
Changing the sign of $x$ in $(-1, 0)$ gives us $(1, 0)$ , which is on the graph!
Same with $(0, 1)$ .
Changing the sign of the x-coordinate gives us $(0, 1)$ , which is already on the graph!

It looks like for every point $(x, y)$ on the graph, the point $(-x, y)$ is also on the graph!
This means the graph is symmetric with respect to the y-axis.

An odd function has the property $f(-x) = -f(x)$ .
This means if we rotate the graph 180 degrees around the origin, it should look exactly the same!

Let's test a point.
We have the point $(1, 0)$ on our graph.
If we change the sign of both the x and y coordinates, we get $(-1, 0)$ .
This point is on the graph!

However, let's try another point.
We have $(2, 1)$ on the graph.
Changing the sign of both coordinates gives us $(-2, -1)$ .
This point is not on the graph!

Since the graph doesn't have symmetry about the origin, it's not an odd function.

The graph is symmetric with respect to the y-axis, but not the origin.
Therefore, the graph represents an even function.