Question
The blue and green points are moveable.
Use the points to transform the graph of `f\left(x\right)=x^{2}`
to the graph of `f\left(x+3\right)`
Hint: Start with the BLUE point and then adjust the GREEN point if needed
Use the points to transform the graph of `f\left(x\right)=x^{2}`
to the graph of `f\left(x+3\right)`
Hint: Start with the BLUE point and then adjust the GREEN point if needed
Answers
GPT-4o mini
To transform the graph of \( f(x) = x^2 \) into the graph of \( f(x + 3) \), we go through a horizontal shift process.
1. **Identify the Effect of the Transformation**:
The transformation from \( f(x) \) to \( f(x + 3) \) indicates we will shift the graph of \( f(x) = x^2 \) to the left by 3 units. The general rule for horizontal shifts is that \( f(x - c) \) shifts the graph to the right by \( c \) units, while \( f(x + c) \) shifts it to the left by \( c \) units.
2. **Move the Blue Point**:
Begin by aligning the blue point, which represents a point on the original graph of \( f(x) = x^2 \), to the position where \( x \) is increased by 3. If you start with the point \( (0, 0) \) on the original graph, you would move it to the left by 3 units, which places it at \( (-3, 0) \).
3. **Adjust the Green Point if Needed**:
The green point should match the corresponding transformed y-coordinate on the new graph. For example, if your blue point was initially \( (0, 0) \) for \( f(0) \), now on the transformed graph \( f(x + 3) \), the new position would be \( (-3, f(-3)) \).
Since \( f(-3) = (-3)^2 = 9 \), you should place the green point at \( (-3, 9) \).
Thus, to summarize:
- Move the blue point from \( (0, 0) \) to \( (-3, 0) \) to reflect the leftward shift.
- Adjust the green point to \( (-3, 9) \) to ensure it represents the correct function value on the new graph of \( f(x + 3) \).
This illustrates the transformation from \( f(x) = x^2 \) to \( f(x + 3) = (x + 3)^2 \).
1. **Identify the Effect of the Transformation**:
The transformation from \( f(x) \) to \( f(x + 3) \) indicates we will shift the graph of \( f(x) = x^2 \) to the left by 3 units. The general rule for horizontal shifts is that \( f(x - c) \) shifts the graph to the right by \( c \) units, while \( f(x + c) \) shifts it to the left by \( c \) units.
2. **Move the Blue Point**:
Begin by aligning the blue point, which represents a point on the original graph of \( f(x) = x^2 \), to the position where \( x \) is increased by 3. If you start with the point \( (0, 0) \) on the original graph, you would move it to the left by 3 units, which places it at \( (-3, 0) \).
3. **Adjust the Green Point if Needed**:
The green point should match the corresponding transformed y-coordinate on the new graph. For example, if your blue point was initially \( (0, 0) \) for \( f(0) \), now on the transformed graph \( f(x + 3) \), the new position would be \( (-3, f(-3)) \).
Since \( f(-3) = (-3)^2 = 9 \), you should place the green point at \( (-3, 9) \).
Thus, to summarize:
- Move the blue point from \( (0, 0) \) to \( (-3, 0) \) to reflect the leftward shift.
- Adjust the green point to \( (-3, 9) \) to ensure it represents the correct function value on the new graph of \( f(x + 3) \).
This illustrates the transformation from \( f(x) = x^2 \) to \( f(x + 3) = (x + 3)^2 \).