Math / Algebra

Question6/16
How did we transform from the parent function? $g(x)=-1 / 5(x-1)^{2}+7$ vertical shrink by $1 / 5$ , left reflection over $x$ -axis, 1, up 7 vertical stretch by $1 / 5$ . left1, up 7 reflection over $X$ -axis, right reflection over $x$ -axis, 7. down 1 vertical shrink by $1 / 5$ , right
1, up 7

Studdy Solution

STEP 1

What is this asking? How does $g(x) = -\frac{1}{5}(x-1)^2 + 7$ transform from the simplest form of this type of function? Watch out! Don't mix up horizontal shifts (left/right) and vertical shifts (up/down)!
Also, remember that the sign of the horizontal shift is the opposite of what it looks like inside the parentheses.

STEP 2

1. Identify the parent function
2. Analyze the transformations

STEP 3

The parent function is the most basic form of the given function.
Think of it as the starting point before any fancy transformations happen.
In our case, the parent function is $f(x) = x^2$ .
It's just a simple parabola!

STEP 4

Let's break down $g(x)$ and see what's different from the parent function.
We have $g(x) = -\frac{1}{5}(x-1)^2 + 7$ .

STEP 5

First, we see a negative sign out front.
This means there's a reflection over the x-axis.
Imagine flipping the parabola upside down!

STEP 6

Next, we have the $\frac{1}{5}$ multiplying the squared term.
Since $\frac{1}{5}$ is less than one, this represents a vertical shrink by a factor of $\frac{1}{5}$ .
The parabola gets compressed vertically, closer to the x-axis.

STEP 7

Now, look inside the parentheses: $(x-1)$ .
This indicates a horizontal shift.
Remember the tricky sign! $(x - 1)$ means a shift 1 unit to the right.

STEP 8

Finally, we have $+ 7$ outside the parentheses.
This represents a vertical shift 7 units up.

STEP 9

The transformations from the parent function $f(x) = x^2$ to $g(x) = -\frac{1}{5}(x-1)^2 + 7$ are: reflection over the x-axis, vertical shrink by a factor of $\frac{1}{5}$ , horizontal shift 1 unit to the right, and vertical shift 7 units up.

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Studdy Solution

STEP 1

STEP 2

1. Identify the parent function
2. Analyze the transformations

STEP 3

The parent function is the most basic form of the given function.
Think of it as the starting point before any fancy transformations happen.
In our case, the parent function is $f(x) = x^2$ .
It's just a simple parabola!

STEP 4

Let's break down $g(x)$ and see what's different from the parent function.
We have $g(x) = -\frac{1}{5}(x-1)^2 + 7$ .

STEP 5

First, we see a negative sign out front.
This means there's a reflection over the x-axis.
Imagine flipping the parabola upside down!

STEP 6

Next, we have the $\frac{1}{5}$ multiplying the squared term.
Since $\frac{1}{5}$ is less than one, this represents a vertical shrink by a factor of $\frac{1}{5}$ .
The parabola gets compressed vertically, closer to the x-axis.

STEP 7

Now, look inside the parentheses: $(x-1)$ .
This indicates a horizontal shift.
Remember the tricky sign! $(x - 1)$ means a shift 1 unit to the right.

STEP 8

Finally, we have $+ 7$ outside the parentheses.
This represents a vertical shift 7 units up.

STEP 9

The transformations from the parent function $f(x) = x^2$ to $g(x) = -\frac{1}{5}(x-1)^2 + 7$ are: reflection over the x-axis, vertical shrink by a factor of $\frac{1}{5}$ , horizontal shift 1 unit to the right, and vertical shift 7 units up.

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Studdy Solution

STEP 1

STEP 2

1. Identify the parent function2. Analyze the transformations

STEP 3

The parent function is the most basic form of the given function.Think of it as the starting point before any fancy transformations happen.In our case, the parent function is f(x)=x2f(x) = x^2f(x)=x2.It's just a simple parabola!

STEP 4

Let's break down g(x)g(x)g(x) and see what's different from the parent function.We have g(x)=−15(x−1)2+7g(x) = -\frac{1}{5}(x-1)^2 + 7g(x)=−51​(x−1)2+7.

STEP 5

First, we see a negative sign out front.This means there's a **reflection** over the x-axis.Imagine flipping the parabola upside down!

STEP 6

Next, we have the 15\frac{1}{5}51​ multiplying the squared term.Since 15\frac{1}{5}51​ is *less than* one, this represents a **vertical shrink** by a factor of 15\frac{1}{5}51​.The parabola gets compressed vertically, closer to the x-axis.

STEP 7

Now, look inside the parentheses: (x−1)(x-1)(x−1).This indicates a **horizontal shift**.Remember the tricky sign! (x−1)(x - 1)(x−1) means a shift **1 unit to the right**.

STEP 8

Finally, we have +7+ 7+7 outside the parentheses.This represents a **vertical shift** **7 units up**.

STEP 9

The transformations from the parent function f(x)=x2f(x) = x^2f(x)=x2 to g(x)=−15(x−1)2+7g(x) = -\frac{1}{5}(x-1)^2 + 7g(x)=−51​(x−1)2+7 are: reflection over the x-axis, vertical shrink by a factor of 15\frac{1}{5}51​, horizontal shift 1 unit to the right, and vertical shift 7 units up.

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Start learning now

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

1. Identify the parent function
2. Analyze the transformations

The parent function is the most basic form of the given function.
Think of it as the starting point before any fancy transformations happen.
In our case, the parent function is $f(x) = x^2$ .
It's just a simple parabola!

Let's break down $g(x)$ and see what's different from the parent function.
We have $g(x) = -\frac{1}{5}(x-1)^2 + 7$ .

First, we see a negative sign out front.
This means there's a reflection over the x-axis.
Imagine flipping the parabola upside down!

Next, we have the $\frac{1}{5}$ multiplying the squared term.
Since $\frac{1}{5}$ is less than one, this represents a vertical shrink by a factor of $\frac{1}{5}$ .
The parabola gets compressed vertically, closer to the x-axis.

Now, look inside the parentheses: $(x-1)$ .
This indicates a horizontal shift.
Remember the tricky sign! $(x - 1)$ means a shift 1 unit to the right.

Finally, we have $+ 7$ outside the parentheses.
This represents a vertical shift 7 units up.

The transformations from the parent function $f(x) = x^2$ to $g(x) = -\frac{1}{5}(x-1)^2 + 7$ are: reflection over the x-axis, vertical shrink by a factor of $\frac{1}{5}$ , horizontal shift 1 unit to the right, and vertical shift 7 units up.