Algebra

Quadratics

Move a curve. Notice its patterns.

Predict, test and explain

Step 1 of 3

Predict

For y = (x − h)² − 4, move h from 0 to 2. Where will the turning point be?

y=x2−4y = x^2 - 4
−6-6
−4-4
−2-2
22
44
66
−10-10
−5-5
55
1010
xx
yy
(−2,  0)(-2,\;0)
(2,  0)(2,\;0)
(0,  −4)(0,\;-4)

Scroll the diagram sideways to see every label.

Turning point(0,  −4)(0,\;-4)Roots(−2,  0),  (2,  0)(-2,\;0),\;(2,\;0)
Transformations and turning points

Tick “Compare with y = x²” to show the starting curve and the turning point’s horizontal and vertical movement.

Start with f(x)=x2f(x)=x^2. Stretch vertically, reflect if needed, then move across and up.

y=af(x−h)+k,a≠0y=a f(x-h)+k,\quad a\ne0

(0,0)⟼(h,k)(0,0)\longmapsto(h,k)

Current transformation: vertical scale factor 1, 0 units right and 4 units down. Turning point: (0,  −4)(0,\;-4)

hh moves the curve across; kk moves it up or down. Changing aa with h,kh,k fixed changes the shape, keeping the turning point fixed.

Predict: what happens to the turning point if you increase hh by one?

InvestigateCan you change the width without moving the turning point?

Explore and compare
Can you break the rule?

Changing only the curve-shape parameter raises the turning point.

y=a(x−h)2+k,a≠0y=a(x-h)^2+k,\quad a\ne0

Always, sometimes or never? Test it in the simulation.

Hint

Choose Graph and Turning point form. Keep h and k fixed, and change a without making it zero.

Check the reasoning

Never. The turning point stays at the same coordinates. Changing a changes the width and the direction of opening.

turning point=(h,k)\text{turning point}=(h,k)

One counterexample disproves “always”. Examples alone do not prove it.

Build a curve with this turning point, passing through the second point.

(h,k)=(2,−3),(0,5)(h,k)=(2,-3),\quad (0,5)

Try it in the simulation. Change one thing at a time.

Hint

Use Graph and Turning point form. Set the turning point first, then adjust the curve shape until the y-intercept is five.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Find the roots of this quadratic.

y=(x−1)2−9y=(x-1)^2-9

Hint

At a root, the curve meets the horizontal axis. Set the output to zero.

Reveal answer

The bracket squared equals nine, so the bracket itself is three or negative three.

x=−2orx=4x=-2\quad\text{or}\quad x=4

Build an explanation

Hint 1

Find where the curve crosses each axis.

Hint 2

In turning-point form, h moves the curve across; k moves it vertically.

Hint 3

For y = a(x − h)² + k, the turning point is (h, k). Set y to zero to find roots.

Worked example

This example uses fixed values, separate from the diagram controls.

Find the turning point and roots of y = (x − 2)² − 9.

  1. Read the turning point from the form: (2, −9).
  2. At a root, y = 0, so (x − 2)² = 9.
  3. Therefore x − 2 = 3 or −3.

Answer: Turning point (2, −9); roots x = −1 and x = 5.

Connect this idea

Watch out: The minus sign inside (x − 2) moves the turning point to x = 2.