Pre-Calculus Chapter 2 Flashcards

Questions and Answers

What is the standard form of a quadratic function?

f(x) = a(x-h)^2 + k, a does not = 0

What is the process of completing the square?

Use (b/2)^2 to find new c.

Add new c and shorten into squared expression.

Subtract new c.

All of the above. (correct)

If a > 0, where does the function have a minimum value?

-b/2a

If a < 0, where does the function have a maximum value?

-b/2a Signup and view all the answers

The graph of a polynomial function is discontinuous.

False Signup and view all the answers

What does the leading coefficient test determine for an odd-degree polynomial with a positive leading coefficient?

Left: Down, Right: Up Signup and view all the answers

What is a function's maximum number of real zeros?

The degree Signup and view all the answers

What is a function's maximum number of relative extrema?

The degree minus 1 Signup and view all the answers

What do repeated zeros indicate in a polynomial?

(x - a)^k, if k is odd the graph crosses the x-axis at x = a, if k is even the graph touches but does not cross. Signup and view all the answers

What is the process of polynomial long division?

Put zeros in as placeholders, divide leading coefficients, multiply, subtract, bring down, repeat. Signup and view all the answers

What is synthetic division?

Can only be used with x-k, switch sign of k, multiply and carry coefficients. Signup and view all the answers

What is the remainder when a polynomial is divided by x-k?

r = f(k) Signup and view all the answers

What does the rational zero test involve?

p/q (factors of constant over factors of leading coefficient) Signup and view all the answers

What does PRZ stand for?

Possible rational zeros Signup and view all the answers

What are complex numbers?

a + bi, includes real and imaginary numbers. Signup and view all the answers

What is the addition formula for complex numbers?

(a + bi) + (c + di) Signup and view all the answers

If f(x) is a polynomial of degree n > 0, what does h have?

n linear factors Signup and view all the answers

If a polynomial has a factor a + bi, what else will it have?

A zero of a - bi Signup and view all the answers

What is the parent function of a rational function?

f(x) = 1/x Signup and view all the answers

How do you find vertical asymptotes?

Find zeros of the denominator Signup and view all the answers

How do you find horizontal asymptotes?

If degree of the numerator = n and degree of the denominator = m, if n < m, there is no horizontal asymptote. Signup and view all the answers

How do you find the x-intercept of a function?

Find zeros of the numerator Signup and view all the answers

How do you find the y-intercept of a function?

Substitute zero for x Signup and view all the answers

What is a slant asymptote?

If the degree of the numerator is exactly 1 more than the degree of the denominator, divide numerator by denominator. Signup and view all the answers

Study Notes

Standard Form of a Quadratic Function

Expressed as f(x) = a(x-h)² + k where a ≠ 0.

Completing the Square

Method to transform a quadratic into vertex form, involving:
- Calculating (b/2)² to find a new constant c.
- Adding this new c, simplifying into a squared expression, then subtracting the new c.

Minimum Value for Quadratics

Occurs at x = -b/(2a) when a > 0.

Maximum Value for Quadratics

Occurs at x = -b/(2a) when a < 0.

Continuity of Polynomial Functions

Graph of polynomial functions is continuous across its entire domain.

Leading Coefficient Test (End Behavior)

Odd Degree, Positive Leading Coefficient: Left down, Right up.
Odd Degree, Negative Leading Coefficient: Left up, Right down.
Even Degree, Positive Leading Coefficient: Left up, Right up.
Even Degree, Negative Leading Coefficient: Left down, Right down.

Maximum Number of Real Zeros in a Function

Determined by the function's degree.

Maximum Number of Relative Extrema

Given by the degree minus one of the polynomial.

Repeated Zeros

Expressed as (x-a)ᶦ, where k > 1.
- If k is odd, the graph crosses the x-axis at x = a.
- If k is even, it touches but does not cross the x-axis at x = a.

Polynomial Long Division

Involves placeholder zeros, matching leading coefficients, multiplying, subtracting, and repeating with subsequent terms.

Synthetic Division

Applicable only for divisors of the form x-k.
Requires switching the sign of k, using coefficients, and sequentially multiplying and adding.

Remainder Theorem for Polynomial Division

The remainder when dividing by x-k is r = f(k).

Rational Zero Test

Identifies potential rational zeros as p/q, where p is factors of the constant and q is factors of the leading coefficient.

Possible Rational Zeros (PRZ)

All combinations of factors of the constant term over factors of the leading coefficient.

Complex Numbers

Form a + bi, combining real numbers (both rational and irrational) and imaginary components.

Addition and Subtraction of Complex Numbers

Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i

Linear Factors of a Polynomial

A polynomial of degree n>0 has exactly n linear factors.

Complex Conjugates

If a polynomial has a factor of a + bi, it necessarily includes a zero of a - bi.

Parent Function of Rational Functions

Represented by f(x) = 1/x.

Finding Vertical Asymptotes

Determined by discovering the zeros of the denominator.

Finding Horizontal Asymptotes

Compare degrees:
- If degree of numerator (n) < degree of denominator (m), asymptote at y = 0.
- If n = m, asymptote found at y = ratio of leading coefficients.
- If n > m, no horizontal asymptote.

Finding X-Intercepts

Derived by finding zeros of the numerator.

Finding Y-Intercept

Found by substituting x = 0 into the function.

Slant Asymptotes

Occur when the degree of the numerator is one more than that of the denominator; found via polynomial division.

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Test your knowledge on quadratic functions with these flashcards from Pre-Calculus Chapter 2. Each card covers essential concepts such as the standard form of a quadratic and methods for finding minimum and maximum values. Perfect for quick revision and deepening your understanding.