Question

1. How can the vertex form of a quadratic function help you sketch the graph of the function?

2. A table of values for the quadratic function g is shown. Do the graphs of the functions g and f(x) = 3(x-1)^2 have the same axis of symmetry ? explain in 8th grade terms
x g(x)
-4 8
-2 3
0 0
6 3

4. Lacretia said the vertex of the function f(x) = (x+2)^2 + 6 is (2, 6). Is she correct? explain your answer

5. graph each function
g(x) = x^2 + 5

Answer 1

1) How the vertex form helps you sketch a quadratic
- Vertex form is y = a(x − h)^2 + k. The point (h, k) is the vertex (the highest or lowest point).
- The line x = h is the axis of symmetry (the parabola is the same on both sides of this line).
- The sign of a tells you whether it opens up (a > 0) or down (a < 0). The size of |a| tells you if it is wider (|a| < 1) or narrower (|a| > 1) than y = x^2.
- To sketch: plot the vertex (h,k), draw the axis x = h, plot a few symmetric points on either side (use x = h ± 1, ±2 etc), then draw the U-shaped curve.

2) Do g and f(x) = 3(x−1)^2 have the same axis of symmetry? (8th-grade explanation)
- The axis of symmetry of f(x) = 3(x − 1)^2 is x = 1 (because the vertex form a(x − h)^2 + k has axis x = h, here h = 1).
- From the table for g:
x: -4 -2 0 6
g(x): 8 3 0 3
Notice g(−2) = 3 and g(6) = 3. Those two x-values are symmetric around the midpoint (−2 + 6)/2 = 2. That means the parabola’s axis is x = 2.
- Therefore g’s axis is x = 2, but f’s axis is x = 1, so they do NOT have the same axis of symmetry.

(Extra small check: you can write g(x) as 0.25(x − 2)^2 − 1 using three points; that shows the vertex is (2, −1) and axis x = 2.)

4) Lacretia’s claim about f(x) = (x + 2)^2 + 6
- Vertex form is a(x − h)^2 + k. Here (x + 2)^2 is the same as (x − (−2))^2, so h = −2 and k = 6.
- The vertex is (−2, 6), not (2, 6). Lacretia is incorrect because she forgot the sign: (x + 2) means the x-coordinate of the vertex is −2.

5) Graphing g(x) = x^2 + 5 (how to plot it)
- This is x^2 shifted up 5. Vertex is (0, 5). Axis of symmetry is x = 0 (the y-axis). It opens upward.
- Useful points to plot:
x = −2 → g = 4 + 5 = 9 → (−2, 9)
x = −1 → g = 1 + 5 = 6 → (−1, 6)
x = 0 → g = 0 + 5 = 5 → (0, 5) (vertex)
x = 1 → g = 1 + 5 = 6 → (1, 6)
x = 2 → g = 4 + 5 = 9 → (2, 9)
- Plot those points, draw a smooth U-shaped curve through them symmetric about x = 0.