Question

do now - mixed review 1
dynamics of algebra 2

1. graph $y = -\frac{1}{3}|x + 2| + 8$
2. write the equation of the graph:
3. graph $y = 2(x - 5)^2 - 6$
4. find the vertex of $y = -3x^2 - 24x - 46$

Explanation:

Problem 4: Find the vertex of \( y = -3x^2 - 24x - 46 \)

Step 1: Recall the vertex formula for a quadratic

For a quadratic function in the form \( y = ax^2 + bx + c \), the x - coordinate of the vertex is given by \( x = -\frac{b}{2a} \).
Here, \( a=-3 \), \( b = - 24 \), and \( c=-46 \).
Substitute \( a \) and \( b \) into the formula:
\( x=-\frac{-24}{2\times(-3)} \)

Step 2: Calculate the x - coordinate

First, simplify the numerator and the denominator:
The numerator \( -(-24)=24 \), the denominator \( 2\times(-3)=-6 \)
Then \( x = \frac{24}{-6}=-4 \)

Step 3: Find the y - coordinate

Substitute \( x = - 4 \) into the original function \( y=-3x^2-24x - 46 \)
\( y=-3\times(-4)^2-24\times(-4)-46 \)
First, calculate \( (-4)^2 = 16 \), then \( -3\times16=-48 \)
\( -24\times(-4) = 96 \)
Now, \( y=-48 + 96-46 \)
\( y=(96-48)-46=48 - 46 = 2 \)

Step 1: Identify the parent function and transformations

The parent function is \( y = |x| \), which has a vertex at \( (0,0) \) and opens upwards.
For the function \( y=-\frac{1}{3}|x + 2|+8 \):

• The \( - \) sign reflects the graph over the x - axis.
• The coefficient \( \frac{1}{3} \) vertically compresses the graph by a factor of \( \frac{1}{3} \).
• The \( x+2 \) inside the absolute value shifts the graph 2 units to the left.
• The \( +8 \) outside the absolute value shifts the graph 8 units up.

Step 2: Find the vertex

The vertex of the absolute - value function \( y=a|x - h|+k \) is at \( (h,k) \). For \( y=-\frac{1}{3}|x + 2|+8 \), we can rewrite it as \( y=-\frac{1}{3}|x-(-2)|+8 \), so the vertex is at \( (-2,8) \)

Step 3: Find two points on each side of the vertex

• For \( x=-2 + 3=-2+3 = 1 \) (we choose a value 3 units to the right of the vertex's x - coordinate to make calculation easy, since the coefficient of the absolute value is \( \frac{1}{3} \)):

\( y=-\frac{1}{3}|1 + 2|+8=-\frac{1}{3}\times3 + 8=-1 + 8 = 7 \)

• For \( x=-2-3=-5 \) (3 units to the left of the vertex's x - coordinate):

\( y=-\frac{1}{3}|-5 + 2|+8=-\frac{1}{3}\times|-3|+8=-\frac{1}{3}\times3 + 8=-1 + 8 = 7 \)

Step 4: Plot the vertex and the points

Plot the vertex \( (-2,8) \), the points \( (1,7) \) and \( (-5,7) \). Then, draw the two rays of the absolute - value graph. The left ray (for \( x\lt - 2 \)) will go from the vertex through \( (-5,7) \) and the right ray (for \( x\gt - 2 \)) will go from the vertex through \( (1,7) \), with the graph opening downwards (because of the negative sign) and vertically compressed.

Problem 3: Graph \( y = 2(x - 5)^2-6 \)

Step 1: Identify the parent function and transformations

The parent function is \( y=x^2 \), which has a vertex at \( (0,0) \) and opens upwards.
For the function \( y = 2(x - 5)^2-6 \):

• The coefficient \( 2 \) vertically stretches the graph by a factor of \( 2 \).
• The \( x - 5 \) inside the square shifts the graph 5 units to the right.
• The \( - 6 \) outside the square shifts the graph 6 units down.

Step 2: Find the vertex

For a quadratic function in the form \( y=a(x - h)^2+k \), the vertex is at \( (h,k) \). For \( y = 2(x - 5)^2-6 \), the vertex is at \( (5,-6) \)

Step 3: Find two points on each side of the vertex

• For \( x=5 + 1=6 \):

\( y=2(6 - 5)^2-6=2\times1^2-6=2 - 6=-4 \)

• For \( x=5-1 = 4 \):

\( y=2(4 - 5)^2-6=2\times(-1)^2-6=2 - 6=-4 \)

Step 4: Plot the vertex and the points

Plot the vertex \( (5,-6) \), the points \( (6,-4) \) and \( (4,-4) \). Then, draw the parabola. Since \( a = 2>0 \), the parabola opens upwards.

Problem 2: Write the equation of the graph (the absolute - value graph)

Answer:

The vertex of the parabola \( y=-3x^2-24x - 46 \) is \( (-4,2) \)

Problem 1: Graph \( y=-\frac{1}{3}|x + 2|+8 \)