Back to the on-screen lesson ·
Read a parabola's direction, zeros, axis of symmetry and vertex from any form of its equation, and say what each means in a situation.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you turn the algebra of quadratics into a picture. You learn what decides which way a parabola opens, how the factored form hands you its crossings, how $-\frac{b}{2a}$ finds the line it is symmetric about, and why the vertex is the answer to every question about a highest or lowest value — a maximum height, a greatest area, a least cost.
You have just factored and solved quadratics, so you can already find the values of $x$ that make one zero. This lesson turns that algebra into a picture: those values are where the curve crosses the axis, and the curve's turning point sits exactly halfway between them.
Parabola: the U-shaped graph of a quadratic function.
Vertex: its turning point — the lowest point if it opens upward, the highest if it opens downward.
Axis of symmetry: the vertical line through the vertex, at $x = -\frac{b}{2a}$. The curve is a mirror image about it.
Vertex form: $y = a(x - h)^2 + k$, with the vertex at $(h, k)$.
Factored form: $y = a(x - r)(x - s)$, with zeros at $r$ and $s$.
A quadratic's graph is a parabola, and which way it opens is decided by the sign of the $x^2$ coefficient alone: positive opens upward with a minimum at the vertex, negative opens downward with a maximum. The factored form $y = a(x - r)(x - s)$ hands you the zeros — where the curve meets the $x$-axis — and because a parabola is symmetric, the axis of symmetry is exactly halfway between them. From the standard form the same line is at $x = -\frac{b}{2a}$, and the vertex's height is whatever the function gives there, so you always find the vertex in two steps: the $x$ first, then substitute. The vertex form $y = a(x - h)^2 + k$ shows the vertex $(h, k)$ outright, which is why completing the square is worth the trouble. In a real situation the vertex is the answer to 'when is it highest' or 'what is the most I can get', and the zeros answer 'when does it hit the ground' or 'when does the profit run out'.
Another way: diagram
A downward parabola with its two $x$-axis crossings marked $-1$ and $3$, a dashed vertical line at $x = 1$ labelled 'axis of symmetry, halfway between the zeros', and the vertex on it labelled 'maximum'.
Another way: story
The three forms are three lenses on one curve. If the question asks where it crosses, look through the factored form; if it asks how high it gets, look through the vertex form; and if you only have the standard form, $-\frac{b}{2a}$ takes you to the vertex in one step.
"$y = (x - 2)^2 - 5$ has vertex $(-2, -5)$." The form already contains a minus sign, so $x - 2$ means $h = 2$: the vertex is $(2, -5)$. Check it by asking which $x$ makes the bracket zero.
"The maximum of $h = -5t^2 + 20t$ is $2$." $t = 2$ is when the maximum happens. The maximum height is $h(2) = 20$. Time and height are different questions.
"A big coefficient of $x$ turns the parabola sideways." Nothing turns a parabola sideways. Only the sign of the $x^2$ coefficient decides up or down; the $x$ term only slides it.
The $x^2$ coefficient is $-2$, which is negative, so the parabola opens downward and the vertex is a maximum.
Sign of $a$ decides direction.
Axis of symmetry: $x = -\frac{b}{2a} = -\frac{4}{2(-2)} = 1$.
The formula gives the $x$ of the vertex.
Height there: $-2(1)^2 + 4(1) + 1 = 3$, so the vertex is $(1, 3)$.
Substitute to get the $y$.
Downward parabola, so the vertex is the maximum.
The $t^2$ coefficient is negative.
$t = -\frac{20}{2(-5)} = 2$ seconds.
When the maximum happens.
$h(2) = -5(4) + 20(2) = -20 + 40 = 20$ metres.
The maximum height itself.
Here $a = 1$ and $b = -8$, so $x = -\frac{-8}{2 \times 1}$.
That gives $x = 4$: the axis of symmetry is the line $x = 4$.
The parabola $y = (x - 2)^2 + 5$ is in vertex form. Fill in its vertex.
vertex $($ x $,$ y $)$
Where does $y = (x + 3)(x - 4)$ cross the $x$-axis? Fill in the two zeros, smaller first.
$x = $ a $\ $ and $\ x = $ b
Which way does $y = -3x^2 - 8x + 4$ open, and what is its vertex?
What is the axis of symmetry of $y = x^2 + 10x + 1$? Give the $x$ value.
answer
A farmer has $72$ m of fencing for three sides of a rectangular pen; the fourth side is an existing wall. If the two sides at right angles to the wall are each $x$ m, what value of $x$ gives the greatest area?
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A ball's height in metres after $t$ seconds is $h = -5t^2 + 30t$. What is its greatest height?
Answer:
You can find a parabola's direction, zeros, axis of symmetry and vertex, and say what they mean. Explain why the vertex of $y = (x - 2)^2 - 5$ is at $(2, -5)$ and not $(-2, -5)$, and why the maximum of $h = -5t^2 + 20t$ is $20$ and not $2$.
8. Your turn: the axis of symmetry of $y = x^2 - 8x + 3$, step 2