Question

on the card below, select which statement is incorrect.

p(x) = -3x² + 12x - 3
#1: p(x) has a vertex at (2, 9).
#2: p(x) passes through the point (-1, -18).
#3: p(x) has a y-intercept at (0, -4).

options: #1, #2, #3

correct and rewrite the incorrect statement.
your answer

Explanation:

Part 1: Identifying the Incorrect Statement

We analyze each statement about the quadratic function \( p(x) = -3x^2 + 12x - 3 \):

Analyzing #1: Vertex at \( (2, 9) \)

For a quadratic function \( ax^2 + bx + c \), the x - coordinate of the vertex is \( x = -\frac{b}{2a} \). Here, \( a=-3 \), \( b = 12 \).
\( x=-\frac{12}{2\times(-3)}=-\frac{12}{-6} = 2 \)
To find the y - coordinate, substitute \( x = 2 \) into \( p(x) \):
\( p(2)=-3(2)^2+12(2)-3=-3\times4 + 24-3=-12 + 24-3=9 \)
So, the vertex is at \( (2, 9) \). Statement #1 is correct.

Analyzing #2: Passes through \( (-1, -18) \)

Substitute \( x=-1 \) into \( p(x) \):
\( p(-1)=-3(-1)^2+12(-1)-3=-3\times1-12 - 3=-3-12 - 3=-18 \)
So, the function passes through \( (-1, -18) \). Statement #2 is correct.

Analyzing #3: y - intercept at \( (0, -4) \)

The y - intercept of a function \( y = p(x) \) is found by substituting \( x = 0 \) into the function.
\( p(0)=-3(0)^2+12(0)-3=0 + 0-3=-3 \)
So, the y - intercept should be at \( (0, -3) \), not \( (0, -4) \). Statement #3 is incorrect.

Part 2: Correcting the Incorrect Statement

The incorrect statement is #3. The correct statement is: \( p(x) \) has a y - intercept at \( (0, -3) \)

Final Answers

• The incorrect statement is \(\boldsymbol{\#3}\)
• The corrected statement: \( p(x) \) has a y - intercept at \( (0, -3) \)

Answer:

Part 1: Identifying the Incorrect Statement

We analyze each statement about the quadratic function \( p(x) = -3x^2 + 12x - 3 \):

Analyzing #1: Vertex at \( (2, 9) \)

For a quadratic function \( ax^2 + bx + c \), the x - coordinate of the vertex is \( x = -\frac{b}{2a} \). Here, \( a=-3 \), \( b = 12 \).
\( x=-\frac{12}{2\times(-3)}=-\frac{12}{-6} = 2 \)
To find the y - coordinate, substitute \( x = 2 \) into \( p(x) \):
\( p(2)=-3(2)^2+12(2)-3=-3\times4 + 24-3=-12 + 24-3=9 \)
So, the vertex is at \( (2, 9) \). Statement #1 is correct.

Analyzing #2: Passes through \( (-1, -18) \)

Substitute \( x=-1 \) into \( p(x) \):
\( p(-1)=-3(-1)^2+12(-1)-3=-3\times1-12 - 3=-3-12 - 3=-18 \)
So, the function passes through \( (-1, -18) \). Statement #2 is correct.

Analyzing #3: y - intercept at \( (0, -4) \)

The y - intercept of a function \( y = p(x) \) is found by substituting \( x = 0 \) into the function.
\( p(0)=-3(0)^2+12(0)-3=0 + 0-3=-3 \)
So, the y - intercept should be at \( (0, -3) \), not \( (0, -4) \). Statement #3 is incorrect.

Part 2: Correcting the Incorrect Statement

The incorrect statement is #3. The correct statement is: \( p(x) \) has a y - intercept at \( (0, -3) \)

Final Answers

• The incorrect statement is \(\boldsymbol{\#3}\)
• The corrected statement: \( p(x) \) has a y - intercept at \( (0, -3) \)