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To determine which of the given points lies on the graph of [tex]\( y = f(x-3) + 5 \)[/tex], we need to understand how the transformation [tex]\( y = f(x-3) + 5 \)[/tex] affects the original point.
Given the original point [tex]\((4, 1)\)[/tex] on the graph of [tex]\( f(x) \)[/tex]:
- [tex]\( x \)[/tex] value of the point is [tex]\( 4 \)[/tex]
- [tex]\( y \)[/tex] value of the point is [tex]\( 1 \)[/tex]
The transformation [tex]\( y = f(x-3) + 5 \)[/tex] consists of:
1. Shifting the graph to the right by 3 units -> [tex]\( x \)[/tex] becomes [tex]\( x - 3 \)[/tex]
2. Shifting the graph up by 5 units -> [tex]\( y \)[/tex] becomes [tex]\( y + 5 \)[/tex]
For point [tex]\((a, b)\)[/tex] to be on the graph of [tex]\( y = f(x-3) + 5 \)[/tex]:
1. The [tex]\( y \)[/tex]-coordinate [tex]\( b \)[/tex] should be 5 units more than the value of [tex]\( f \)[/tex] at [tex]\( (a-3) \)[/tex]
2. The [tex]\( x \)[/tex]-coordinate [tex]\( a-3 \)[/tex] should correspond to the [tex]\( x \)[/tex]-coordinate of the original point.
So, let's check each option:
1. Checking [tex]\((-3, 5)\)[/tex]:
- [tex]\( a = -3 \)[/tex]
- [tex]\( b = 5 \)[/tex]
- The original [tex]\( x \)[/tex]-value would be [tex]\( -3 - 3 = -6 \)[/tex]
- The new [tex]\( y \)[/tex]-value would need to be [tex]\( 5 - 5 = 0 \)[/tex]
- [tex]\((-6, 0)\)[/tex] is not the original point [tex]\((4, 1)\)[/tex], so [tex]\((-3, 5)\)[/tex] is not on the transformed graph.
2. Checking [tex]\((4, 9)\)[/tex]:
- [tex]\( a = 4 \)[/tex]
- [tex]\( b = 9 \)[/tex]
- The original [tex]\( x \)[/tex]-value would be [tex]\( 4 - 3 = 1 \)[/tex]
- The new [tex]\( y \)[/tex]-value would need to be [tex]\( 9 - 5 = 4 \)[/tex]
- [tex]\((1, 4)\)[/tex] is not the original point [tex]\((4, 1)\)[/tex], so [tex]\((4, 9)\)[/tex] is not on the transformed graph.
3. Checking [tex]\((7, 6)\)[/tex]:
- [tex]\( a = 7 \)[/tex]
- [tex]\( b = 6 \)[/tex]
- The original [tex]\( x \)[/tex]-value would be [tex]\( 7 - 3 = 4 \)[/tex]
- The new [tex]\( y \)[/tex]-value would need to be [tex]\( 6 - 5 = 1 \)[/tex]
- [tex]\((4, 1)\)[/tex] is the original point, so [tex]\((7, 6)\)[/tex] is on the transformed graph.
4. Checking [tex]\((1, 6)\)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 6 \)[/tex]
- The original [tex]\( x \)[/tex]-value would be [tex]\( 1 - 3 = -2 \)[/tex]
- The new [tex]\( y \)[/tex]-value would need to be [tex]\( 6 - 5 = 1 \)[/tex]
- [tex]\((-2, 1)\)[/tex] is not the original point [tex]\((4, 1)\)[/tex], so [tex]\((1, 6)\)[/tex] is not on the transformed graph.
Therefore, the point [tex]\((7, 6)\)[/tex] must also be on the graph of [tex]\( y = f(x-3) + 5 \)[/tex].
To visualize this:
- Original point: [tex]\((4, 1)\)[/tex]
- Transformed point: [tex]\((7, 6)\)[/tex]
Both points are represented as follows on the coordinate plane:
- Original point: [tex]\((4, 1)\)[/tex]
- Transformed point: [tex]\((7, 6)\)[/tex]
Given the original point [tex]\((4, 1)\)[/tex] on the graph of [tex]\( f(x) \)[/tex]:
- [tex]\( x \)[/tex] value of the point is [tex]\( 4 \)[/tex]
- [tex]\( y \)[/tex] value of the point is [tex]\( 1 \)[/tex]
The transformation [tex]\( y = f(x-3) + 5 \)[/tex] consists of:
1. Shifting the graph to the right by 3 units -> [tex]\( x \)[/tex] becomes [tex]\( x - 3 \)[/tex]
2. Shifting the graph up by 5 units -> [tex]\( y \)[/tex] becomes [tex]\( y + 5 \)[/tex]
For point [tex]\((a, b)\)[/tex] to be on the graph of [tex]\( y = f(x-3) + 5 \)[/tex]:
1. The [tex]\( y \)[/tex]-coordinate [tex]\( b \)[/tex] should be 5 units more than the value of [tex]\( f \)[/tex] at [tex]\( (a-3) \)[/tex]
2. The [tex]\( x \)[/tex]-coordinate [tex]\( a-3 \)[/tex] should correspond to the [tex]\( x \)[/tex]-coordinate of the original point.
So, let's check each option:
1. Checking [tex]\((-3, 5)\)[/tex]:
- [tex]\( a = -3 \)[/tex]
- [tex]\( b = 5 \)[/tex]
- The original [tex]\( x \)[/tex]-value would be [tex]\( -3 - 3 = -6 \)[/tex]
- The new [tex]\( y \)[/tex]-value would need to be [tex]\( 5 - 5 = 0 \)[/tex]
- [tex]\((-6, 0)\)[/tex] is not the original point [tex]\((4, 1)\)[/tex], so [tex]\((-3, 5)\)[/tex] is not on the transformed graph.
2. Checking [tex]\((4, 9)\)[/tex]:
- [tex]\( a = 4 \)[/tex]
- [tex]\( b = 9 \)[/tex]
- The original [tex]\( x \)[/tex]-value would be [tex]\( 4 - 3 = 1 \)[/tex]
- The new [tex]\( y \)[/tex]-value would need to be [tex]\( 9 - 5 = 4 \)[/tex]
- [tex]\((1, 4)\)[/tex] is not the original point [tex]\((4, 1)\)[/tex], so [tex]\((4, 9)\)[/tex] is not on the transformed graph.
3. Checking [tex]\((7, 6)\)[/tex]:
- [tex]\( a = 7 \)[/tex]
- [tex]\( b = 6 \)[/tex]
- The original [tex]\( x \)[/tex]-value would be [tex]\( 7 - 3 = 4 \)[/tex]
- The new [tex]\( y \)[/tex]-value would need to be [tex]\( 6 - 5 = 1 \)[/tex]
- [tex]\((4, 1)\)[/tex] is the original point, so [tex]\((7, 6)\)[/tex] is on the transformed graph.
4. Checking [tex]\((1, 6)\)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 6 \)[/tex]
- The original [tex]\( x \)[/tex]-value would be [tex]\( 1 - 3 = -2 \)[/tex]
- The new [tex]\( y \)[/tex]-value would need to be [tex]\( 6 - 5 = 1 \)[/tex]
- [tex]\((-2, 1)\)[/tex] is not the original point [tex]\((4, 1)\)[/tex], so [tex]\((1, 6)\)[/tex] is not on the transformed graph.
Therefore, the point [tex]\((7, 6)\)[/tex] must also be on the graph of [tex]\( y = f(x-3) + 5 \)[/tex].
To visualize this:
- Original point: [tex]\((4, 1)\)[/tex]
- Transformed point: [tex]\((7, 6)\)[/tex]
Both points are represented as follows on the coordinate plane:
- Original point: [tex]\((4, 1)\)[/tex]
- Transformed point: [tex]\((7, 6)\)[/tex]
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