Quadratics

Question 1

What is a quadratic function?

Answer 1

f(x) = ax^2 + bx + c, where a is not equal to 0

Question 2

What is a parabola?

Answer 2

The graph of a quadratic function, U-shaped

Question 3

What do congruent parabolas imply?

Answer 3

They have the same stretch factor |a|, but differ in position

Question 4

State the key features of a parabola

Answer 4

Direction of opening, min/max, vertex, axis of symmetry, y-int, x-int/roots/zeros

Question 5

Describe the direction of opening of a parabola

Answer 5

If a > 0, it opens upwards and creates a minimum value
If a < 0, it opens downwards and creates a maximum value

Question 6

Describe the vertex of a parabola. How do you find the vertex from different forms?

Answer 6

The highest or lowest point of the parabola.
From vertex form: (h, k) is the vertex
From standard form: Use x = -b/2a, then y = f(x) to find the vertex

Question 7

What is x = -b/2a, where does it come from?

Answer 7

It’s the x-coordinate of the vertex defined by the standard quadratic equation. Completing the square of this form derives the formula.

Question 8

Describe the axis of symmetry of a parabola

Answer 8

A vertical line that passes through the vertex. The equation is x = h or x = -b/2a

Question 9

Describe the y-int of a parabola

Answer 9

Substitute x = 0 to find the y-int.
From standard form: (0,c) is the y-int

Question 10

Describe the x-ints/roots/zeros of a parabola

Answer 10

From factored form: r1 and r2
From standard form: Use the quadratic formula or factor to find the roots

Question 11

What’s the equation for standard form and its best use?

Answer 11

f(x) = ax^2 + bx + c
Best for finding y-int and quadratic formula utilization

Question 12

What’s the equation for vertex form and its best use?

Answer 12

f(x) = a(x - h)^2 + k
Best for graphing and transformations

Question 13

What’s the equation for factored form and its best use?

Answer 13

f(x) = a(x - r1)(x - r2)
Best for finding x-ints/roots

Question 14

How do you convert vertex to standard form?

Answer 14

Expand

Question 15

How do you convert standard to factored form?

Answer 15

Factor or quadratic formula

Question 16

How do you convert standard to vertex form?

Answer 16

Complete the square

Question 17

How do you complete the square?

Answer 17

Practice. Example in notes

Question 18

What are the methods for solving quadratics

Answer 18

Solving a quadratic means finding its roots/x-ints that make the equation true.
Factoring, quadratic formula, graphing are methods.

Question 19

How do you graph an equation in standard form?

Answer 19

Convert it to vertex or factored form, then graph

Question 20

How do you graph an equation in vertex form?

Answer 20

(h, k) is the vertex, the 1st point
1. Move 1 unit to the left & right of the vertex. The y value is +- (1^2)(|a|)
2. Move 2 units to the left & right of the vertex. The y value is +- (2^2)(|a|)
3. Repeat if needed. Need at least 5 points

Question 21

How do you graph an equation in factored form?

Answer 21

r1 & r2 are x-ints/roots. Vertex is (h, k)
1. h = (r1 + r2) / 2
2. k = (h - r1)(h - r2)
3. Need at least 3 points

Question 22

What is the discriminant (D) ?

Answer 22

A part of the quadratic formula that determines the type and number of solutions for a quadratic equation.

Question 23

Describe the 3 cases of discriminant values

Answer 23

If D > 0: 2 real solutions
If D = 0: 1 real solution / 2 repeated real solutions
If D < 0: no real solutions / 2 imaginary solutions

Question 24

What is the sum & product of the roots of a quadratic?

Answer 24

For ax^2 + bx + c = 0, the sum is -b/a and the product is c/a

For x^2 - (m+n)x + mn, the sum is m + n and the product is mn. This is the factored and expanded version of the standard equation.

Question 25

What's a quadratic inequality?

Answer 25

A quadratic inequality is a mathematical statement that compares a quadratic expression to another value using inequality symbols, asking for the range of variables that make the statement true

Question 26

How do you solve a quadratic inequality?

Answer 26

1. Find the roots of the quadratic 2. Graph the quadratic 3. Determine where the parabola satisfies the inequality. Use interval notation

Question 27

Explain intersection of a line and a parabola applications

Answer 27

To find the POI coordinate, equate the equations of the line & parabola to find x-values. Then substitute those values to find the y-values. To find the number of solutions, equate the equations to form a quadratic, then use the discriminant