Today's lesson
We're starting Chapter 7 — Parabolas. A parabola is the smooth U-shaped curve you get when you graph a quadratic rule. Today is all about the simplest one, $y=x^2$, and how changing the rule moves and reshapes it.
Learning intentions
- Know the shape and symmetry of the basic parabola $y=x^2$
- Identify the key features of a parabola from its graph: turning point, axis of symmetry, maximum/minimum, $x$- and $y$-intercepts
- See the effect of transforming $y=x^2$ — dilation (wider/narrower), reflection ($a<0$) and translation
- Build a table of values and use it to plot a parabola accurately
Part 1 — The basic parabola $y=x^2$ (~8 min)
Start by building a table of values for $y=x^2$, then plot the points. Because squaring a negative gives a positive, the left side mirrors the right side — the curve is symmetric.
| $x$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|---|---|---|
| $y=x^2$ | $9$ | $4$ | $1$ | $0$ | $1$ | $4$ | $9$ |
Key ideas — the basic parabola $y=x^2$
- The vertex (turning point) is at $(0,0)$ and it is a minimum.
- The axis of symmetry is the vertical line $x=0$ (the $y$-axis).
- The $y$-intercept is $(0,0)$ and the $x$-intercept is $(0,0)$.
- Transformations of $y=x^2$ include dilation (wider/narrower), reflection in the $x$-axis (when $a<0$, the parabola opens downward) and translation (sliding the curve).
📺 Walkthrough: how the number $a$ in $y=ax^2$ stretches or reflects the parabola, and where the vertex and axis of symmetry stay put.
Part 2 — Reading key features (Example 1, ~12 min)
From a graph you should be able to read off four things: the turning point (and whether it's a max or min), the axis of symmetry, the $x$-intercepts and the $y$-intercept.
- i Turning point is a minimum at $(1,-4)$ (the curve opens up).
- ii Axis of symmetry is the vertical line through the vertex: $x=1$.
- iii $x$-intercepts (where the curve crosses the $x$-axis): $(-1,0)$ and $(3,0)$.
- iv $y$-intercept (where it crosses the $y$-axis): $(0,-3)$.
Now you try: A parabola has a maximum turning point at $(-2,0)$, passes through $(0,-4)$ and opens downward. State its axis of symmetry and $y$-intercept. Answer: axis of symmetry $x=-2$; $y$-intercept $(0,-4)$; only one $x$-intercept, at $(-2,0)$.
Building understanding — read each feature off the graph above (Example 1).
- Is the turning point a maximum or a minimum?
- Write the coordinates of the turning point.
- Write the $y$-intercept coordinates.
- Write both $x$-intercepts.
- State the axis of symmetry.
Part 3 — Transforming $y=x^2$ (Example 2, ~12 min)
The number $a$ in $y=ax^2$ controls width and direction; adding/subtracting and bracketing slides the curve. For each rule below we describe whether it's a max or min, whether it's reflected, its turning point, the $y$-value when $x=1$, and whether it's wider or narrower than $y=x^2$.
| Rule | Max / min | Reflected? | Turning point | $y$ at $x=1$ | vs $y=x^2$ |
|---|---|---|---|---|---|
| $y=4x^2$ | minimum | no | $(0,0)$ | $4$ | narrower |
| $y=(x+2)^2$ | minimum | no | $(-2,0)$ | $9$ | same |
| $y=-x^2+3$ | maximum | yes | $(0,3)$ | $2$ | same |
Now you try: Complete the same table for $y=\tfrac12 x^2$, $y=(x-2)^2$ and $y=-x^2-1$. Answers: $\tfrac12 x^2$ → min, no, $(0,0)$, $y=\tfrac12$, wider; $(x-2)^2$ → min, no, $(2,0)$, $y=1$, same; $-x^2-1$ → max, yes, $(0,-1)$, $y=-2$, same.
Practice 3.1 — describe each parabola (max/min, reflected?, turning point, wider/narrower/same).
- $y=3x^2$
- $y=-2x^2$
- $y=\tfrac14 x^2$
- $y=(x-1)^2$
- $y=x^2+2$
- $y=-x^2+5$
b) max, yes, $(0,0)$, narrower
c) min, no, $(0,0)$, wider
d) min, no, $(1,0)$, same
e) min, no, $(0,2)$, same
f) max, yes, $(0,5)$, same
Part 4 — Plotting from a table of values (~8 min)
The surest way to know your graph is correct is to build a table first. Substitute each $x$-value into the rule, then plot the points and join them with a smooth curve.
| $x$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|---|---|---|
| $y=2x^2$ | $18$ | $8$ | $2$ | $0$ | $2$ | $8$ | $18$ |
| $y=x^2-3$ | $6$ | $1$ | $-2$ | $-3$ | $-2$ | $1$ | $6$ |
Check: $y=2x^2$ is narrower than $y=x^2$ (values grow twice as fast); $y=x^2-3$ is $y=x^2$ slid down 3, so its vertex is $(0,-3)$.
Practice 4.1 — complete the table of values (use $-3\le x\le 3$).
| $x$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|---|---|---|
| $y=-x^2$ | fill in | ||||||
| $y=(x-2)^2$ | fill in | ||||||
$y=(x-2)^2$: $25,\ 16,\ 9,\ 4,\ 1,\ 0,\ 1$ (vertex at $(2,0)$)
Part 5 — Quick quiz (5 min)
Pick the correct answer for each, then click Mark.
Q1. The turning point of the basic parabola $y=x^2$ is at:
Q2. Which parabola is narrower than $y=x^2$?
Q3. The graph of $y=-x^2$ is the graph of $y=x^2$:
Q4. A parabola has its minimum turning point at $(1,-4)$. Its axis of symmetry is:
Q5. For $y=x^2-3$, the $y$-value when $x=2$ is:
Working program — Cambridge Ex 7A (p590)
After the quiz, open Cambridge Chapter 7 (page 590) and complete the following:
| Set work | Extension |
|---|---|
| Questions 1–7 (½), 8–11 | Questions 12, 13 (½), 14 |
"(½)" means do every second part. Show a table of values for any graphing question.
Exit ticket — write in your book
- What are the coordinates of the turning point of $y=x^2$, and is it a maximum or a minimum?
- What does a negative value of $a$ do to the graph of $y=ax^2$?
- How can you tell from a graph where the axis of symmetry is?