Question

stained glass window project linear & quadratic equations
directions
step 1. in the table below, circle 3 equations each from groups a, b, c, d, e and f. you will have a total of 18 equations circled.
step 2. record each equation you pick on on your data sheet pages. you must record them in order, equation #1 - 3 from group a, equation #4 - 6 from group b, etc
step 3. fill - in/complete the data sheet pages for all 18 of your equations. be sure to show some work for equations from groups c, d, e and f.
step 4. graph all 18 equations on your stained glass window graph paper. every equation must be labeled with its equation number from your data sheets
step 5. color/design your stained glass window.
group a group b group c group d group e group f
x = 4 y = - 8 - 3y + x = 18 3y + 3x = 9 y = \frac{1}{2}(x - 6)^2 - 7 y = -\frac{1}{2}(x + 6)^2 + 6
x = - 2 y = - 5 2y = 4x - 2y - x = 10 y = \frac{1}{4}(x + 2)^2 - 10 y = -\frac{1}{3}(x - 3)^2 + 3
x = 7 y = 2 y - 3x = 9 2y + 6x = 4 y = \frac{1}{4}(x + 2)^2 - 10 y = -\frac{1}{3}(x - 3)^2 + 3
x = - 6 y = 6 2y - x = 2 y + 2x = 8 y = \frac{1}{3}(x - 3)^2 - 2 y = -\frac{1}{2}(x + 6)^2 - 2
x = 1 y = 8 3y - 2x = 21 3y + x = - 3 y = \frac{1}{2}(x + 4)^2 + 1 y = -\frac{1}{4}(x - 2)^2 + 9
x = 8 y = - 2 2y + 6 = 2x 3y + 2x = - 6 y = \frac{1}{2}(x + 4)^2 + 1 y = -\frac{1}{4}(x - 2)^2 + 9
x = 3 y = 7 2y - 3x = - 16 - 2y = x - 12 y = \frac{1}{3}(x + 3)^2 - 5 y = -\frac{1}{3}(x + 3)^2 + 10

Explanation:

Step1: Follow step - 1 of the directions

From group A, we can choose \(x = 4\), \(x=-2\), \(x = 7\). From group B, we can choose \(y=-8\), \(y=-5\), \(y = 2\). From group C, we can choose \(-3y + x=18\), \(2y = 4x\), \(y-3x = 9\). From group D, we can choose \(3y + 3x=9\), \(-2y-x = 10\), \(2y+6x = 4\). From group E, we can choose \(y=\frac{1}{2}(x - 6)^2-7\), \(y=\frac{1}{4}(x + 2)^2-10\), \(y=\frac{1}{3}(x - 3)^2-2\). From group F, we can choose \(y=-\frac{1}{2}(x + 6)^2+6\), \(y=-\frac{1}{3}(x - 3)^2+3\), \(y=-\frac{1}{2}(x + 6)^2-2\).

Step2: Follow step - 2 of the directions

Record these equations in order on the data - sheet pages as per the requirement. For example, equations 1 - 3 are from group A, 4 - 6 from group B and so on.

Step3: Follow step - 3 of the directions

For linear equations (groups A, B, C, D), we can rewrite them in slope - intercept form \(y=mx + b\) (if not already in that form). For example, for \(-3y + x=18\), we solve for \(y\):
\[

$$\begin{align*} -3y&=-x + 18\\ y&=\frac{1}{3}x-6 \end{align*}$$

\]
For quadratic equations (groups E, F), we can identify the vertex form \(y=a(x - h)^2+k\), where \((h,k)\) is the vertex. For \(y=\frac{1}{2}(x - 6)^2-7\), the vertex is \((6,-7)\) and \(a=\frac{1}{2}\).

Step4: Follow step - 4 of the directions

Graph the linear equations using the slope and y - intercept (or x - intercept for vertical lines like \(x = 4\)) and graph the quadratic equations using the vertex and the shape determined by the value of \(a\). Label each equation with its corresponding number from the data - sheets.

Step5: Follow step - 5 of the directions

Color/design the stained - glass window as per personal preference.

Since no specific question about solving or graphing a particular equation is asked, we have provided a general guideline on how to complete the project as per the given steps. If we assume a question like "Graph the equation \(y=\frac{1}{2}(x - 6)^2-7\)":

Step1: Identify the vertex

The vertex form of a quadratic equation is \(y=a(x - h)^2+k\). For \(y=\frac{1}{2}(x - 6)^2-7\), \(h = 6\) and \(k=-7\), so the vertex is \((6,-7)\).

Step2: Determine the shape

Since \(a=\frac{1}{2}>0\), the parabola opens upwards.

Step3: Find additional points

Choose some \(x\) - values, for example, when \(x = 4\):
\[

$$\begin{align*} y&=\frac{1}{2}(4 - 6)^2-7\\ &=\frac{1}{2}(-2)^2-7\\ &=\frac{1}{2}\times4-7\\ &=2 - 7\\ &=-5 \end{align*}$$

\]
When \(x = 8\):
\[

$$\begin{align*} y&=\frac{1}{2}(8 - 6)^2-7\\ &=\frac{1}{2}(2)^2-7\\ &=2 - 7\\ &=-5 \end{align*}$$

\]
Plot the vertex \((6,-7)\) and the points \((4,-5)\) and \((8,-5)\) and draw a smooth curve to form the parabola.

Answer:

The steps above show how to complete the stained - glass window project and how to graph a sample quadratic equation. For a full answer, one would need to follow all the steps for all 18 chosen equations.