The graph of the quadratic function is shown in the coordinate plane.
Which point lies on the graph of ?
When a function is transformed, track one clear point from the original graph (often the vertex or an intercept). For , change the -coordinate so that the input to stays the same (this shifts points left 1). Then apply the outside operations to the old -value: multiplying by a number scales (and possibly reflects) vertically, and adding a constant shifts vertically.
Hints
Use an easy-to-read point
Pick a point on the graph that has clear integer coordinates, like the vertex or an intercept.
Interpret
For , think about how the -coordinate of a point changes so that the input to stays the same.
Transform the -value
After handling the horizontal shift, apply the outside operations to the old -value in the order: multiply by , then add .
Desmos Guide
Enter the quadratic from the graph
Using the x-intercepts and and the y-intercept , type
and adjust the slider until the graph matches the given parabola (it will match when ).
Graph the transformed function
In a new line, type
(or, if you set , you can type ).
Check the answer choices as points
Type each answer choice as a point, like , and see which point lies on the transformed graph.
Step-by-step Explanation
Read a key point from the graph
From the graph of , the vertex is at , so .
Account for the horizontal shift
In , the input to is .
To use the point where the input equals 2, set , so .
Apply the vertical transformations
Now compute the new -value at :
So the point on the transformed graph is .