Transform a quadratic and find its key features

On the solid graph y=2x², what is the output when x=1?

Investigate y=a(x−h)²+k. First keep h=k=0 and try a=2, 0.5, −1 and 0. Then restore a=1 and move h alone, then k alone. Finally combine changes. V marks (h,k); P is one unit to its right. When a=0, the solid graph is a horizontal line and V is no longer a turning point. At a=1, h=k=0, the curves coincide

For nonzero a, which statement agrees with these experiments?

Change one parameter at a time

A parameter is a number in a rule that we vary while holding the others fixed. Compare tables and graphs on the same axes and scales. Record a prediction, several trials and a conclusion with an example

Vertical stretch and compression

Multiply each output of the base curve by the coefficient. If the coefficient is positive, the graph still opens upward. A factor greater than 1 stretches vertically and makes the parabola narrower. A factor between 0 and 1 compresses vertically and makes it wider

Rule

\(\displaystyle y=\frac12x^2\)

For the compressed graph, find y when x=2.

Square 2 first, then multiply by one half

Which graph is wider than y=x² on the same axes?

Negative coefficients

A negative a reflects the vertically scaled base graph in the x-axis. The sign gives the opening direction; its magnitude gives the stretch or compression. Reflection alone does not change width

Describe y=−½x² relative to y=x²

Rule

\(\displaystyle y=-2x^2\)

At x=2, what is the output of the reflected, stretched rule?

Square x before multiplying by −2.

The zero boundary

At a=0, the squared term disappears. The graph is the horizontal line y=k. It has no unique turning point and is not a parabola. Our quadratic practice uses a≠0.

Horizontal and vertical translation

The squared part is zero at x=h, so y=k there. For a≠0 the vertex is (h,k). In x−h, a positive h moves right and a negative h moves left. Adding k moves the graph up when k>0 and down when k<0.

Rule

\(\displaystyle y=2(x+3)^2-4\)

Read the horizontal coordinate of the vertex

Which x-value makes x+3 zero?

Rule

\(\displaystyle y=2(x+3)^2-4\)

Read the vertical coordinate of the same vertex

The number added outside the squared term is k.

Which equation is the axis of symmetry for y=2(x+3)²−4?

Order matters

Starting with y=x², reflect and vertically stretch or compress first. Then translate horizontally by h and vertically by k. Translating vertically before scaling generally gives a different final height

Which ordered description produces y=−2(x−1)²+3?

Check transformed points

A point on the base curve transforms as shown. Use a vertex and a symmetric pair to check a sketch. This also explains why the marked point is the vertex

Rule

\(\displaystyle y=-2(x-1)^2+3\)

For y=−2(x−1)²+3, find the output at x=2.

Evaluate 2−1, square it, multiply by −2, then add 3.

A student says the vertex of y=−2(x−1)²+3 is a minimum. Is that right?

Count intercepts without a new formula

Set y=0 and isolate the square. A square equal to a negative number has no real solution; equal to zero gives one distinct solution; equal to a positive number gives two. This method does not need the quadratic formula or discriminant

How many distinct x-intercepts does y=2(x+1)²+3 have?

How many distinct x-intercepts does y=−2(x−3)² have?

Keep both square roots

Set the output to zero and isolate the square. Take both signs to find the roots shown

Rule

\(\displaystyle y=-\frac12(x-2)^2+8\)

For the rule shown, enter the smaller x-intercept

Set y=0, isolate the square and keep both signs

Independent parameter investigation

Choose three nonzero a-values including a fraction and a negative value. Keep h and k fixed. Predict and then graph each rule using a digital tool; record the vertex and a symmetric pair of points. Next vary h alone and k alone. Explain the changes using your tables and graphs. Include a=0 as a boundary case. Ask a teacher or peer to review the predictions, labelled axes, consistent scales, coordinates and conclusions