Question

math 2 chapter 9 review
name mia ambriz per. 4th date jun 21,

1. write the equation for a parabola that has x-intercepts at (0,-2) and (0,6) and a = -1.

standard form: factored form: vertex form:

1. write the equation for a parabola that has its vertex at (-6, -32) and a = 2.

standard form: factored form: vertex form:

1. solve for x using at least 2 different methods. 3x² - 6x - 20 = 4

1. consider the equation y = x².write the new equation that:

• shifts the parabola 2 units to the right
• reflects the parabola across the x-axis
• stretches the parabola vertically by a factor of 3.
• and shifts the parabola up 4 units

Explanation:

Problem 1

Factored Form

Step1: Recall factored form of parabola

The factored form of a parabola is \( y = a(x - r_1)(x - r_2) \), where \( r_1 \) and \( r_2 \) are the x - intercepts. Here, \( r_1=-2 \), \( r_2 = 6 \) and \( a=-1 \).

Step2: Substitute values into factored form

Substitute \( a=-1 \), \( r_1=-2 \), \( r_2 = 6 \) into the formula: \( y=-1(x - (-2))(x - 6)=-(x + 2)(x - 6) \)

Vertex Form

Step1: Find the vertex's x - coordinate

The x - coordinate of the vertex of a parabola with x - intercepts \( x_1=-2 \) and \( x_2 = 6 \) is given by \( x=\frac{x_1 + x_2}{2}=\frac{-2 + 6}{2}=2 \)

Step2: Find the y - coordinate of the vertex

Substitute \( x = 2 \) into the factored form \( y=-(x + 2)(x - 6) \). \( y=-(2 + 2)(2 - 6)=-4\times(-4) = 16 \). So the vertex is \( (2,16) \)

Step3: Write vertex form

The vertex form of a parabola is \( y=a(x - h)^2+k \), where \( (h,k) \) is the vertex. Here \( a=-1 \), \( h = 2 \), \( k = 16 \). So \( y=-1(x - 2)^2+16=-(x - 2)^2+16 \)

Standard Form

Step1: Expand the factored form

Start with \( y=-(x + 2)(x - 6) \). First, multiply \( (x + 2)(x - 6)=x^2-6x+2x - 12=x^2-4x - 12 \)

Step2: Apply the negative sign

\( y=-(x^2-4x - 12)=-x^2 + 4x+12 \)

Problem 2

Vertex Form

Answer:

s

Problem 1

• Factored form: \( \boldsymbol{y=-(x + 2)(x - 6)} \)
• Vertex form: \( \boldsymbol{y=-(x - 2)^2+16} \)
• Standard form: \( \boldsymbol{y=-x^2 + 4x+12} \)

Problem 2

• Vertex form: \( \boldsymbol{y=2(x + 6)^2-32} \)
• Standard form: \( \boldsymbol{y=2x^2+24x + 40} \)
• Factored form: \( \boldsymbol{y=2(x + 10)(x + 2)} \)

Problem 3

• Solutions: \( \boldsymbol{x = 4} \) or \( \boldsymbol{x=-2} \)

Problem 4

• Equation: \( \boldsymbol{y=-3(x - 2)^2+4} \)