MISSION · Algebra
Reading Quadratic Functions
Read the story told by a parabola and its equation
LEVEL 1 · EXPLORE
Make the idea move
Change the controls and watch what happens. Follow your curiosity and see what you notice.
Loading the activity…
LEVEL 2 · LEARN
Unlock the explanation
Open the chapters in any order. Try the examples as you go.
The idea in one sentence
A quadratic function graphs as a parabola, and its form can reveal the turning point, direction, and zeros.
Start with the parent function
The simplest quadratic is:
\[y = x^2\]Its graph is a U-shaped curve with vertex \((0,0)\). Because the coefficient of \(x^2\) is positive, it opens upward.
Read vertex form
Vertex form is:
\[y = a(x-h)^2 + k\]The vertex is \((h,k)\). For \(y = 2(x-3)^2 - 4\):
- the vertex is \((3,-4)\);
- the parabola opens upward because \(a=2\) is positive;
-
it is narrower than \(y=x^2\) because $$ a >1$$.
Watch the sign inside the parentheses: \((x-3)\) gives \(h=3\), while \((x+3)\) gives \(h=-3\).
Find the zeros
Zeros are the \(x\)-values where \(y=0\). If a quadratic is factored:
\[y=(x-2)(x+5)\]Set each factor equal to zero. The zeros are \(x=2\) and \(x=-5\). Those are the points where the graph crosses the \(x\)-axis.
Common mix-ups
- Reading the vertex sign incorrectly in vertex form.
- Assuming every parabola crosses the \(x\)-axis twice. It may cross twice, touch once, or never cross.
- Looking only at \(a\) for the vertex. The coefficient controls direction and width, not location.
Check yourself
For \(y=-3(x+1)^2+6\), name the vertex and say whether the parabola opens up or down.
The vertex is \((-1,6)\), and the parabola opens down because \(a=-3\).
LEVEL 3 · REFLECT
Add this quest to your trail
When you feel ready, mark this idea as explored. You can always come back and try it again.
EXPLORER STRATEGIES