Activity 8: Quadratic Recognition Speed Drill

Objective

Train yourself to identify the three most important features of a quadratic without graphing, factoring, completing the square, or using a calculator.

For each equation, answer:

Question A: Does the parabola open up or down?

Question B: How many distinct real roots/solutions does it have?

Question C: What is the exact y-intercept?

How to Answer Instantly

Question A: Up or Down?

Look only at the leading coefficient.

Positive:

$a>0$

→ Opens Up

Negative:

$a<0$

→ Opens Down

Question B: Number of Real Roots

Use whichever clue is easiest.

Factored Form:

$y=a(x-r_1)(x-r_2)$

Two different factors → 2 roots
Same factor twice → 1 root

Vertex Form:

$y=a(x-h)^2+k$

(using Desmos) Ask:

Does the vertex touch the x-axis? → 1 root
Does it cross the x-axis? → 2 roots
Does it miss the x-axis? → 0 roots

Standard Form:

$y=ax^2+bx+c$

Use the discriminant:

$b^2-4ac$

$>0$ → 2 roots
$=0$ → 1 root
$<0$ → 0 roots

Question C: Y-Intercept

Set:

$x=0$

Whatever y becomes is the y-intercept.

Write the answer as:

$(0,y)$

Drill Table

#	Quadratic Equation	Opens Up/Down	# of Real Roots	Y-Intercept
1	$y=3x^2-5x-2$
2	$y=-2x^2+4x-5$
3	$y=x^2-8x+16$
4	$y=-0.5(x-3)^2+8$
5	$y=4(x+2)(x-5)$
6	$y=-x^2-6x-9$
7	$y=2x^2+7x+10$
8	$y=-(x+4)^2$
9	$y=5x^2-20$
10	$y=-3x^2+2x$
11	$y=(x-1)^2+4$
12	$y=-2(x+5)(x+1)$
13	$y=x^2+x+1$
14	$y=-4x^2+12x-9$
15	$y=0.1x^2-10$
16	$y=2(x-6)^2-2$
17	$y=-7x^2-14x-7$
18	$y=3x(x-8)$
19	$y=-x^2+5x-8$
20	$y=12x^2+5x-3$

‣

Drill Table: Answer Key (toggle)

#	Quadratic Equation	Opens Up/Down	# of Real Roots	Y-Intercept
1	$y=3x^2-5x-2$	Up	2	$(0,-2)$
2	$y=-2x^2+4x-5$	Down	0	$(0,-5)$
3	$y=x^2-8x+16$	Up	1	$(0,16)$
4	$y=-0.5(x-3)^2+8$	Down	2	$(0,3.5)$
5	$y=4(x+2)(x-5)$	Up	2	$(0,-40)$
6	$y=-x^2-6x-9$	Down	1	$(0,-9)$
7	$y=2x^2+7x+10$	Up	0	$(0,10)$
8	$y=-(x+4)^2$	Down	1	$(0,-16)$
9	$y=5x^2-20$	Up	2	$(0,-20)$
10	$y=-3x^2+2x$	Down	2	$(0,0)$
11	$y=(x-1)^2+4$	Up	0	$(0,5)$
12	$y=-2(x+5)(x+1)$	Down	2	$(0,-10)$
13	$y=x^2+x+1$	Up	0	$(0,1)$
14	$y=-4x^2+12x-9$	Down	1	$(0,-9)$
15	$y=0.1x^2-10$	Up	2	$(0,-10)$
16	$y=2(x-6)^2-2$	Up	2	$(0,70)$
17	$y=-7x^2-14x-7$	Down	1	$(0,-7)$
18	$y=3x(x-8)$	Up	2	$(0,0)$
19	$y=-x^2+5x-8$	Down	0	$(0,-8)$
20	$y=12x^2+5x-3$	Up	2	$(0,-3)$

SAT Reflection Questions

After completing the drill, answer:

Which form was easiest to identify roots from?

Standard
Vertex
Factored

Which form was easiest to identify the y-intercept from?
Which form was easiest to identify whether the graph opened up or down?
Which problems required the discriminant?
Which problems could be solved without any algebra at all?

The goal is not computation.

The goal is recognition.