QuestionExplain the relationship between the graphs of $g(x) = x^{2} + 1$ and $f(x) = x^{2}$ .

Studdy Solution

STEP 1

Assumptions1. We are comparing the graphs of two functions, $g(x) = x^{} +1$ and $f(x) = x^{}$ . . We are looking for the relationship between these two graphs.

STEP 2

First, let's consider the function $f(x) = x^{2}$ . This is a standard quadratic function, which forms a parabola with its vertex at the origin (0,0) and opens upwards.

STEP 3

Now, let's consider the function $g(x) = x^{2} +1$ . This is also a quadratic function, but it is a transformation of the function $f(x) = x^{2}$ .

STEP 4

The transformation from $f(x) = x^{2}$ to $g(x) = x^{2} +1$ is a vertical shift upwards by1 unit. This is because the "+1" in the function $g(x) = x^{2} +1$ adds1 to the output of the function $f(x) = x^{2}$ for every input $x$ .

STEP 5

In terms of the graph, this means that every point $(x, y)$ on the graph of $f(x) = x^{2}$ is shifted to a new point $(x, y+1)$ on the graph of $g(x) = x^{2} +1$ .

STEP 6

Therefore, the graph of $g(x) = x^{2} +1$ is the same as the graph of $f(x) = x^{2}$ , but shifted1 unit upwards.
The relationship between the graph of the function $g(x) = x^{2} +1$ and the graph of the function $f(x) = x^{2}$ is that the graph of $g(x)$ is the graph of $f(x)$ shifted1 unit upwards.

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QuestionExplain the relationship between the graphs of $g(x) = x^{2} + 1$ and $f(x) = x^{2}$ .

Studdy Solution

STEP 1

Assumptions1. We are comparing the graphs of two functions, $g(x) = x^{} +1$ and $f(x) = x^{}$ . . We are looking for the relationship between these two graphs.

STEP 2

First, let's consider the function $f(x) = x^{2}$ . This is a standard quadratic function, which forms a parabola with its vertex at the origin (0,0) and opens upwards.

STEP 3

Now, let's consider the function $g(x) = x^{2} +1$ . This is also a quadratic function, but it is a transformation of the function $f(x) = x^{2}$ .

STEP 4

The transformation from $f(x) = x^{2}$ to $g(x) = x^{2} +1$ is a vertical shift upwards by1 unit. This is because the "+1" in the function $g(x) = x^{2} +1$ adds1 to the output of the function $f(x) = x^{2}$ for every input $x$ .

STEP 5

In terms of the graph, this means that every point $(x, y)$ on the graph of $f(x) = x^{2}$ is shifted to a new point $(x, y+1)$ on the graph of $g(x) = x^{2} +1$ .

STEP 6

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QuestionExplain the relationship between the graphs of g(x)=x2+1g(x) = x^{2} + 1g(x)=x2+1 and f(x)=x2f(x) = x^{2}f(x)=x2.

Studdy Solution

STEP 1

Assumptions1. We are comparing the graphs of two functions, g(x)=x+1g(x) = x^{} +1g(x)=x+1 and f(x)=xf(x) = x^{}f(x)=x. . We are looking for the relationship between these two graphs.

STEP 2

First, let's consider the function f(x)=x2f(x) = x^{2}f(x)=x2. This is a standard quadratic function, which forms a parabola with its vertex at the origin (0,0) and opens upwards.

STEP 3

Now, let's consider the function g(x)=x2+1g(x) = x^{2} +1g(x)=x2+1. This is also a quadratic function, but it is a transformation of the function f(x)=x2f(x) = x^{2}f(x)=x2.

STEP 4

The transformation from f(x)=x2f(x) = x^{2}f(x)=x2 to g(x)=x2+1g(x) = x^{2} +1g(x)=x2+1 is a vertical shift upwards by1 unit. This is because the "+1" in the function g(x)=x2+1g(x) = x^{2} +1g(x)=x2+1 adds1 to the output of the function f(x)=x2f(x) = x^{2}f(x)=x2 for every input xxx.

STEP 5

In terms of the graph, this means that every point (x,y)(x, y)(x,y) on the graph of f(x)=x2f(x) = x^{2}f(x)=x2 is shifted to a new point (x,y+1)(x, y+1)(x,y+1) on the graph of g(x)=x2+1g(x) = x^{2} +1g(x)=x2+1.

STEP 6

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QuestionExplain the relationship between the graphs of $g(x) = x^{2} + 1$ and $f(x) = x^{2}$ .

Assumptions1. We are comparing the graphs of two functions, $g(x) = x^{} +1$ and $f(x) = x^{}$ . . We are looking for the relationship between these two graphs.

First, let's consider the function $f(x) = x^{2}$ . This is a standard quadratic function, which forms a parabola with its vertex at the origin (0,0) and opens upwards.

Now, let's consider the function $g(x) = x^{2} +1$ . This is also a quadratic function, but it is a transformation of the function $f(x) = x^{2}$ .

The transformation from $f(x) = x^{2}$ to $g(x) = x^{2} +1$ is a vertical shift upwards by1 unit. This is because the "+1" in the function $g(x) = x^{2} +1$ adds1 to the output of the function $f(x) = x^{2}$ for every input $x$ .

In terms of the graph, this means that every point $(x, y)$ on the graph of $f(x) = x^{2}$ is shifted to a new point $(x, y+1)$ on the graph of $g(x) = x^{2} +1$ .