Lesson 2.3: Quadratic Functions: Graphs & Features

Parabolas, Vertices, and Symmetry

Understand y = ax² + bx + c and y = a(x - h)² + k. Determine axis of symmetry, vertex, zeros, and whether the vertex is a max or min.

Learning Objectives

Identify standard and vertex forms
Find axis of symmetry and vertex
Determine zeros and intercepts
Interpret max/min and context

Forms & Features

Standard form: $y=ax^2+bx+c$ . Axis: $x=-\dfrac{b}{2a}$ .

Vertex form: $y=a(x-h)^2+k$ . Vertex at $(h,\ k)$ . If $a>0$ open up (min), if $a<0$ open down (max).

Completing square (derive vertex form): $y=ax^2+bx+c=a\Big(x^2+\tfrac{b}{a}x\Big)+c$ , add and subtract $(\tfrac{b}{2a})^2$ : $y=a\Big(x+\tfrac{b}{2a}\Big)^2+\Big(c-\tfrac{b^2}{4a}\Big)$ → vertex $(h,k)=(-\tfrac{b}{2a},\ c-\tfrac{b^2}{4a})$ .

Zeros & discriminant: Solve $ax^2+bx+c=0$ → roots depend on $\Delta=b^2-4ac$ : $\Delta>0$ two real zeros, $\Delta=0$ one real (double) zero, $\Delta<0$ no real zeros.

Intercepts: y-intercept at $(0,c)$ ; x-intercepts are roots of $ax^2+bx+c=0$ when they exist.

Advanced Worked Examples

A) Convert to Vertex Form & Identify Features

Given $y=3x^2-12x+7$ , write in vertex form; find axis, vertex, min value, and y-intercept.

Complete square: $y=3\big[x^2-4x\big]+7=3\big[(x-2)^2-4\big]+7=3(x-2)^2-12+7=3(x-2)^2-5$

Axis: $x=2$ ; Vertex: $(2,-5)$ ; min value $-5$ (since $a=3>0$ ).

y-intercept: set $x=0$ → $y=7$ → $(0,7)$

B) Find Zeros & Factor

For $y=x^2-5x+6$ , find zeros and intercepts; factor if possible.

Factor: $x^2-5x+6=(x-2)(x-3)$

Zeros: $x=2,\ 3$ → x-intercepts $(2,0),(3,0)$ ; y-intercept $(0,6)$

C) Projectile Height Model

A ball is launched: $h(t)=-16t^2+32t+5$ (feet). Find time to reach max height and the max height.

Axis $t=-\tfrac{b}{2a}=\tfrac{-32}{2(-16)}=1$ s

Max height $h(1)=-16+32+5=21$ ft

D) Discriminant Insight

For $y=2x^2+4x+5$ , determine whether the graph crosses the x-axis.

$\Delta=b^2-4ac=16-40=-24<0$ → no real zeros → no x-intercepts; always above x-axis since $a>0$ and vertex y > 0.

Common Pitfalls

Forgetting to factor out $a$ before completing the square when $a\ne 1$ .
Misreading axis formula sign: axis is $x=-\tfrac{b}{2a}$ , not $\tfrac{b}{2a}$ .
Assuming two zeros always: check discriminant; $\Delta\le 0$ changes the count.
Confusing vertex y with y-intercept; compute separately.

Practice

1) Write $y=2x^2-8x+3$ in vertex form; give vertex and min value.

Solution

Factor 2:

y=2[x^2-4x]+3=2[(x-2)^2-4]+3=2(x-2)^2-8+3=2(x-2)^2-5

→ vertex (2, -5), min -5

2) For $y=-x^2+6x-5$ , find zeros and max value.

Solution

Complete square:

y=-(x^2-6x)+-5=-(x-3)^2+9-5=-(x-3)^2+4

→ vertex (3,4), max 4; zeros at

x=1,5

3) A revenue model: $R(p)=-200p^2+4000p$ . Find price p maximizing revenue and the max revenue.

Solution

Axis:

p=-\tfrac{b}{2a}=\tfrac{-4000}{-400}=10

;

R(10)= -200\cdot100 + 4000\cdot10 = -20000 + 40000 = 20000

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