Download to graph quadratics in standard form Facts/Formulas: A quadratic

Ch. 1 Quadratics
1.1 Quadratics in Standard Form
A. OBJ: to graph quadratics in standard form
B.Facts/Formulas:
1. A quadratic function is a function that can be
written in the standard form y = ax2+bx+c where a ≠
0. The graph of a quadratic equation is a parabola.
2. The lowest or highest point on a parabola is the
vertex.
3. The axis of symmetry divides the parabola into
mirror images and passes through the vertex.
4. For y = ax2+bx+c, the vertex’s y-coordinate is the
minimum value of the function if a > 0 and the
maximum value if a < 0.
5. a>0 up and a< 0 down
6. |a| > 1 narrow and |a| < 1 wide
7. axis of symmetry =
−𝒃
𝟐𝒂
8. y – intercept = (0, c)
1.2 Quadratics in Intercept and Vertex Form
A.OBJ: to graph quads in intercept and vertex form
B. FACTS/FORMULAS:
1. The vertex form of a quadratic equation is given by
y = a(x - h)2 +k.
2. The intercept form of a quadratic equation is given
by
y = a(x - p)(x - q).
C. Examples:
1.
1.3 / 1.4 Solving Quadratics
A.OBJ: to solve quads by factoring
B.Facts/Formulas:
1. A monomial is an expression that is a number, a
variable, or the product of a number and one or more
variables.
2. A binomial is the sum of two monomials.
3. A trinomial is the sum of three monomials.
4. A quadratic equation in one variable can be written as
ax2 + bx + c = 0 where a ≠ 0.
5. A solution of a quadratic equation is called the root or
solution of the equation.
6. Because a function’s value is zero when x = p and x =
q, the numbers, p and q are also called the zeros or xintercepts of the function.
7. Zero Product Property: If A*B = 0, then A=0 or
B=0.
C. Examples:
1.5 Solving by √
A. OBJ: to solve quads by using square roots
B.Facts/Formulas:
1. A number r is a square root of a number s if r2= s.
2. The expression √𝒔 is called a radical. The symbol √
is a radical sign and the number s beneath the radical sign
is the radicand of the expression.
3. To rationalize a denominator √𝒃 of a fraction, multiply
the numerator and denominator by √𝒃
4. To rationalize a denominator a + √𝒃 of a fraction, multiply
the numerator and denominator by a - √𝒃 , and to
rationalize a denominator a - √𝒃 of a fraction, multiply
the numerator and denominator by a + √𝒃
5. The expressions a + √𝒃 and a - √𝒃 are called conjugates.
1.6 Complex #’s
A. OBJ: to perform operations on complex numbers.
B. Facts/Formulas:
1. The imaginary unit i is defined as i = √−𝟏
2. A complex number written in standard form is a
number a ± bi, where a and b are real numbers. If b ≠ 0,
then a ± bi is an imaginary number.
3. Two complex numbers of the form a + bi and a - bi are
called complex conjugates.
4. Every complex number corresponds to a point in the
complex plane.
5. The complex plane has a horizontal axis called the real
axis and a vertical axis called the imaginary axis.
6. The absolute value of a complex number z = a ± bi,
denoted |z| is a nonnegative real number defined as
z = √𝒂𝟐 + 𝒃𝟐 .
1.7 Completing the Square
A. OBJ: to solve quads by completing the square.
B.Facts/Formulas:
1. To complete the square for ax2+bx=c, add (b/2)2
to both sides.
2. Remember that a = 1!
3. Factor and solve
C. Ex:
1.8 Quadratic Formula
A. OBJ: to solve using the quadratic formula.
B.Facts/Formulas:
1. The quadratic formula: Let a, b, and c be real numbers
where a ≠ 0.
2. The solutions of the quadratic equation ax2 ± bx ± c = 0
−𝑏±√𝑏2 −4𝑎𝑐
are 𝑥 =
2𝑎
3. In the quadratic formula, the expression b2 - 4ac is called
the discriminant of the quadratic equation.
a. IF the discriminant > 0, then there are 2 real
solutions
b. IF the discriminant < 0, then there are 2 imaginary
solutions
c. IF the discriminant = 0, then there is only 1 real
solution.
1.9 Quadratic Inequalities
A. OBJ: to solve and graph quadratic inequalities
B.Facts/Formulas:
1. A quadratic inequality in two variables can be
written in one of the following forms:
y < ax2 ±bx ± c
y ax2 ± bx±c
y ≥ ax2 ± bx ± c
y ≤ax2 ± bx ± c
2. A quadratic inequality in one variable can be
written in one of the following forms:
ax2 ± bx ± c < 0
ax2 ± bx ± c 0
ax2 ± bx ± c ≥ 0
ax2 ± bx ± c ≤0
C. Ex: