integrated math

What is the vertex form of a quadratic equation?

y = a(x – h)² + k

In vertex form, what does the (h, k) represent?

The vertex of the parabola

What does a negative "a" value do to the graph of a quadratic?

It reflects (flips) the parabola over the x-axis

What transformation happens in y = (x + 4)²?

The graph shifts 4 units left

What transformation happens in y = (x – 2)² + 5?

➡ Moves right 2 (because of the –2 inside the parentheses)
➡ Goes up 5 (because of the +5 outside)

🧠 think of it like this:

• the number inside the (x – #) moves the graph side to side, but it’s kinda sneaky—it does the opposite of what it looks like

• the number outside (like +5) just moves it up or down, no tricks

What does the "a" value in vertex form control?

The width of the parabola and whether it opens up or down

What happens to the graph of y = 3(x – 1)² – 2?

➡ Moves right 1 (because of –1)
➡ Goes down 2 (because of –2)
➡ Gets skinnier (because the 3 makes it stretch vertically)

🧠 remember:

• bigger numbers in front = narrower graph (like it’s getting squeezed)

• smaller fractions = wider (like it’s chillin’ and spreading out)

What transformation does the equation f(x) = |x – 4| + 2 represent?
🅐 Shift 4 units left, 2 units up
🅑 Shift 4 units right, 2 units up
🅒 Shift 4 units right, 2 units down
🅓 Reflect over x-axis, shift up 2

🅑 Shift 4 units right, 2 units up

True or False:
The graph of f(x) = –|x + 3| opens downward and is shifted 3 units to the left.

A: True

What happens when “a” is greater than 1 in a quadratic function?

The graph gets narrower (vertical stretch)

What happens when “a” is between 0 and 1 (like 0.5)?

The graph gets wider (vertical compression)

How does f(x) = 0.5(x – 2)² + 5 compare to f(x) = x²?

• Wider (compressed by 0.5)

• Shift right 2

• Shift up 5

Write a quadratic that is: reflected, stretched by 3, shifted left 4 and up 2.

f(x) = –3(x + 4)² + 2

Given the vertex form equation for a quadratic:
  f(x) = a(x − h)² + k
If I wanted to shift my parabola up, I would...

increase the value of k
(because k controls the vertical shift — more k = graph goes up ⬆)

Which of the following transformations occurs when you change the function from f(x)=x to f(x)=x−4?

This transformation shifts the graph of the function 4 units to the right.

the y axis is in what direction

up and down