**Middle School Number Sense Lesson 104: Parabolas**

(I'd like to give a shout out to

**Jin**for teaching the teacher!)

Until recently, this concept only showed up on high school tests. This year, it appeared

**7 times**on middle school tests, with a median placement at

**question # 75**.

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A parabola will have an equation in the form:

**f(x) = Ax2 + Bx + C**

Its vertex will be at point

**(h, k)**, where: h =

**-B/2A**

**, and**k = f(h) =

**f(-B/2A)**

These problems will often ask for

**h or k**, but they also may ask for elements of

**transformations**(shifts, dilations, reflections, etc.). Two helpful websites to review are:

- https://www.purplemath.com/modules/fcntrans.htm
- https://www.whitman.edu/mathematics/calculus_online/section01.04.html

**Rules of translation/transformation:**

**f(x) + b**shifts the function**b**units**upward****f(x) – b**shifts the function**b**units**downward****f(x + b)**shifts the function**b**units to the**left****f(x – b)**shifts the function**b**units to the**right**- -
**f(x)**reflects the function in the x-axis (so upside-down) **f(-x)**reflects the function in the y-axis (swapping the left & right sides)

**Example 1:**

**If f(x) = 3x**

^{2}- 18x + 21 and g(x) = f(x - 4), then g(x) has an axis of symmetry of x = ___- The axis of symmetry of f(x) is x = h.
- h = -B/2A = -(-18)/(2)(3) = 18/6 =
**3** - f(x – 4) means we shift the function 4 units to the right. 3 + 4 =
**7**

Example 2:

Example 2:

**f(x) = 2x**

^{2}- 24x + 7 and g(x - 3), then g(x) has an axis of symmetry of x = ___- The axis of symmetry of f(x) is x = h.
- H = -B/2A = -(-24)/(2)(2) = 24/4 =
**6** - f(x -3) means we shift 3 units to the right. 6 + 3 =
**9**

****

**Example 3: f(x) is a parabola with vertex (2, -4). g(x) = 5f(x – 4) + 11. g(x) has a vertex of (h, k). h = ___**

- We are worried only about h, which is
**x – 4**(forget about the 5 and the +11) - x – 4 means we shift the vertex 4 units to the right. 2 + 4 =
**6**

**Example 4: f(x) is a quadratic with a vertex of (2, 13). 2f(x – 4) + 7 has a vertex of (h, k) and k = ___**

- We are not worried about (x – 4). We have a vertical stretch by a factor of 2, and a vertical shift up 7.
- 2(13) + 7 = 26 + 7 =
**33**

**Example 5: f(x) is a parabola with vertex (3, -4). g(x) = 2f(x – 4). G(x) has a vertex of (h, k). h + k = ___**

**h**: shifted 4 units to the right. 3 + 4 =**7****k**: vertical stretch by a factor of 2. -4 x 2**=**-**8**- h + k = 7 + (-8) =
**-1**

**Up Next for Middle School: SubRev3Digit**