Every quadratic ax2+bx+c graphs as a parabola — a U-shape (when a>0) or upside-down U (when a<0). To sketch it cleanly, find four landmarks: the vertex, the axis of symmetry, the y-intercept, and the x-intercepts.
In our function: a=1 (positive → opens up), b=−4, c=3.
Every quadratic ax2+bx+c graphs as a parabola — a U-shape (when a>0) or upside-down U (when a<0). To sketch it cleanly, find four landmarks: the vertex, the axis of symmetry, the y-intercept, and the x-intercepts.
In our function: a=1 (positive → opens up), b=−4, c=3.
Step 02 of 05
Step 1 — axis of symmetry and vertex. The axis is the vertical line through the vertex:
x=−2ab
x=−2(1)−4=2
So the axis is x=2. To get the vertex's y-value, plug x=2 back into f:
f(2)=(2)2−4(2)+3=4−8+3=−1
Vertex: (2,−1).
Step 03 of 05
Step 2 — y-intercept. Set x=0; the constant c IS the y-intercept.
f(0)=0−0+3=3
y-intercept: (0,3). Free landmark — no work required beyond reading off c.
Step 04 of 05
Step 3 — x-intercepts. Set f(x)=0 and solve. Factor first if you can:
x2−4x+3=0
(x−1)(x−3)=0
x-intercepts: (1,0) and (3,0). They flank the vertex symmetrically — both are 1 unit from x=2.
Step 05 of 05
Step 4 — sketch. Plot all four landmarks, draw the axis of symmetry, mirror the y-intercept across it, then draw a smooth U through the lot.
The grey point (4,3) is the y-intercept mirrored across the axis.
Key insight
Four landmarks pin down a parabola: vertex, axis of symmetry, y-intercept, x-intercepts. x=−b/(2a) gets you the axis; everything else falls out from substitution or factoring.
A1.K.2 — Vertex form and transformations.
Vertex form
f(x)=a(x−h)2+k
Step 01 of 05
Vertex form bakes the vertex coordinates straight into the formula. Given f(x)=a(x−h)2+k:
Vertex
→(h,k)
Axis of symmetry
→x=h
Opens up if
→a>0
Opens down if
→a<0
No factoring, no −b/(2a) formula. Read the vertex right off the equation.
Step 02 of 05
Sign trap on h. Vertex form has a MINUS: (x−h)2. So:
(x−3)2+5
→vertex (h,k)=(3,5)
(x+3)2+5
→vertex (−3,5) (because x+3=x−(−3))
(x−4)2−7
→vertex (4,−7)
h flips sign reading off the equation; k reads as written.
Step 03 of 05
Transformations from the parenty=x2. Each parameter shifts or stretches the basic parabola:
h
→horizontal shift: right by h, left if negative
k
→vertical shift: up by k, down if negative
a>1
→vertical stretch (narrower)
0<a<1
→vertical compress (wider)
a<0
→flips upside down (opens down)
Step 04 of 05
Worked example. Describe the graph of:
f(x)=−2(x+1)2+8
Read off the parameters:
a=−2
→opens down, vertical stretch by 2
h=−1
→shifted LEFT 1 (because of the +1 inside)
k=8
→shifted UP 8
Vertex
→(−1,8)
Dashed grey: parent y=x2. Solid: transformed.
Step 05 of 05
When you'd CONVERT to vertex form. If you start with ax2+bx+c form, complete the square to get vertex form. Or just use x=−b/(2a) for the vertex if you only need the vertex itself — same answer, less algebra.
Key insight
Standard form is best for finding intercepts. Vertex form is best for finding the vertex and describing transformations. Pick the form that matches the question — and remember the h sign flip.
A1.K.3 — Maximum and minimum value.
The function
f(x)=−2x2+12x−5
Step 01 of 05
The vertex of a parabola IS the maximum or minimum value of the function.
a>0 (opens up)
→vertex is the minimum
a<0 (opens down)
→vertex is the maximum
Our function has a=−2 (negative) → opens down → vertex is the MAXIMUM.
Step 02 of 05
Step 1 — find the vertex's x-coordinate.
x=−2ab=−2(−2)12=−−412=3
The maximum occurs AT x=3.
Step 03 of 05
Step 2 — find the vertex's y-value by plugging back in.
f(3)=−2(3)2+12(3)−5=−18+36−5=13
Vertex: (3,13). The maximum VALUE of the function is 13; it's reached when x=3.
Step 04 of 05
Two different questions, two different answers.
"What is the max value?"
→13 (the y-coordinate)
"At what x is the max reached?"
→3 (the x-coordinate)
"What is the maximum point?"
→(3,13) (both)
Pay attention to which one the problem actually asks.
Step 05 of 05
Word-problem reading. Many real-world quadratics describe height, profit, or area where the max/min is the answer.
Projectile height
→h(t)=−16t2+v0t+h0 — opens down, vertex = max height
Revenue
→price × quantity often quadratic, vertex = optimal price
Area with fixed perimeter
→vertex = max area
Key insight
"Maximum value" or "minimum value" of a quadratic = the y-coordinate of the vertex. Find x=−b/(2a), plug back in, that's it. The sign of a tells you which one (max if down-open, min if up-open).
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