SP #1: Unit E Concept 1 - Graphing a quadratic and identifying all key parts

*Notice- the scale on the graph was modified for the function to fit. There is color coding to distinguish: blue- parent graph; yellow- vertex; orange- y intercept; purple- axis of symmetry: and pink- x intercepts.

Finding the Parent Graph: Solving 3x^2-18x-22=5

1. Add 22 to both sides and factor out a 3. 

2 a. Use the vertex form equation [(b/2)^2] to create a prefect square, which will end up with 9. 

b. 9 will be added to both sides in the boxes and the prefect square can be completed on the right hand side: 3(x-3)^2. 

c. Add/multiply the numbers on the left to get 54.

3. The equation 3(x-3)^2=54 is what is left. Subtract 54 to get the parent graph: f(x)= 3(x-3)^2-54

Solving 3x^2-18x-22=5 for X Intercept/s 

*Do the previous 1-2 steps.

.4. The equation left off was 3(x-3)^2=54. Solve for x algebraically to find the x intercepts. There will be three possibilities: two answers, one answer, or none (imaginary numbers). Ex: (answer value, 0)

Make the original standard from equation equal to 0 and solve to find the y intercept. The vertex will be (h,k) from the parent graph and the axis of symmetry will be x= "h".

This problem is about sketching quadratics in standard form (f(x)= ax^2+ bx+ c) by converting it into parent form (f(x)= a(x-h)^2+k) for simplicity. In order to do so, one must fully complete the square. With the parent graph, you'll need to find the vertex, y-intercept, axis of symmetry, and x-intercepts in order to graphs. Several algebraic steps will be needed to be processed to find few of the requirements.

Pay special attention that (h,k) from the parent function form are the (x,y) values of the vertex. For "k", simply shift up or down with the corresponding number value. For "h" make x-h=0 and solve (note: the value from h should be the opposite from the original value) Also remember that the quadratic function will have two, one, or none (imaginary numbers) x intercepts. And thirdly, if "a" is (+) positive, then the graph will have a minimum, since the vertex will be on the bottom of the curve. If "a" is (-), then the graph will have a maximum, since the vertex will be on the top of the curve. With the parent form, the sketches will be more accurate and detailed.