Question

a student used the steps shown to graph ( f(x) = (x - 1)^2 + 6 ). describe and correct the student’s error.
1 plot the vertex at ( (-1, 6) ).
2 graph points at ( (-2, 15) ) and ( (-3, 22) ).
3 reflect the points across the axis of symmetry ( x = 1 ).
4 connect the points with a parabola.

describe and correct the student’s error. select the correct choice below and fill in the answer box to complete your choice

a. the axis of symmetry is incorrect. the axis of symmetry should be (square).
(type an equation.)

b. the graphed point ( (-2, 15) ) is incorrect. when ( x = -2 ), the value of ( f(x) ) is (square).

c. the graphed point ( (-3, 22) ) is incorrect. when ( x = -3 ), the value of ( f(x) ) is (square).

d. the vertex is incorrect. the vertex should be plotted at (square).

Explanation:

Step1: Recall vertex form of parabola

The vertex form of a parabola is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \( x = h \) is the axis of symmetry.
For \( f(x)=(x - 1)^2+6 \), comparing with \( a(x - h)^2 + k \), we have \( h = 1 \) and \( k = 6 \). So the vertex should be \((1, 6)\), not \((- 1,6)\).

Step2: Analyze other options (optional but for clarity)

• Option A: The axis of symmetry for \( f(x)=(x - 1)^2+6 \) is \( x = 1 \), so this is correct.
• Option B: Let's calculate \( f(-2)\). Substitute \( x=-2 \) into \( f(x)=(x - 1)^2+6 \), we get \( f(-2)=(-2 - 1)^2+6=(-3)^2 + 6=9 + 6 = 15 \). So this point is correct.
• Option C: Calculate \( f(-3)\). Substitute \( x = -3 \) into \( f(x)=(x - 1)^2+6 \), we get \( f(-3)=(-3 - 1)^2+6=(-4)^2+6 = 16 + 6=22 \). So this point is correct.

Answer:

D. The vertex is incorrect. The vertex should be plotted at \((1, 6)\)