A given vertex, the parabola’s turning point, is a cue to use vertex form: . The vertex fixes and , while another point fixes , which controls the parabola’s opening and width. Use a Desmos regression for the point condition. Watch the sign inside the square: the square must be zero at the vertex’s input.
Hints
- Hint 1
In vertex form, , the square becomes zero when . Which squared expression becomes zero at the given vertex input?
- Hint 2
A point on a function’s graph means : put its first coordinate into the function and set the result equal to its second coordinate. That leaves only to find.
Step-by-step
Use vertex form and the extra point
Step 1Build the form from the vertex
In vertex form, , the vertex is because the square is zero at . Here the square must be zero at , so use , not . The vertex output is , so . Only is unknown.
- Step 2
Turn the other point into an equation
A point on the graph gives an input and its output. The point means , so substitute for and for : . Use this point, rather than the vertex, because the vertex makes the squared term zero and cannot determine .
- Step 3
Find the missing coefficient in Desmos
Type . The tells Desmos to find the unknown coefficient that makes the point condition true. Under PARAMETERS, Desmos shows .
- Step 4
Write the equation that meets both conditions
Replace with in the vertex form: . Its square is zero at , giving the vertex output ; the regression made its output at equal . It meets both given conditions. Choice D.