Precalculus Formulas Flashcards
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Questions and Answers

What is the Slope-Intercept Form?

  • I = Prt
  • y-y1=m(x-x1)
  • d = rt
  • y = mx + b (correct)

What is the Slope-Point Form?

y - y1 = m(x - x1)

What is the Slope Formula?

(y₂ - y₁) / (x₂ - x₁)

What does L₁ | L₂ represent?

<p>m₁ = -1 / m₂</p> Signup and view all the answers

What is the Speed-Time-Distance formula?

<p>d = rt</p> Signup and view all the answers

What is the formula for Simple Interest?

<p>I = Prt</p> Signup and view all the answers

What is the formula for Compounded Interest?

<p>A = P(1 + r/n)^(nt)</p> Signup and view all the answers

What is the Carbon-14 Decay Rate?

<p>-0.00012</p> Signup and view all the answers

What is the formula for Interest Compounded Continuously?

<p>A = Pe^(rt)</p> Signup and view all the answers

What is the Half-Life Formula?

<p>1/2 = e^(rt)</p> Signup and view all the answers

What is the Doubling Time Formula?

<p>2 = e^(rt)</p> Signup and view all the answers

What is the Distance Formula?

<p>d = √[(x₂ - x₁)² + (y₂ - y₁)²]</p> Signup and view all the answers

What is the Midpoint Formula?

<p>(x₁ + x₂)/2, (y₁ + y₂)/2</p> Signup and view all the answers

What is the Circle Formula?

<p>(x - h)² + (y - k)² = r²</p> Signup and view all the answers

What is a Complex Number?

<p>a + bi</p> Signup and view all the answers

What is the Difference Quotient?

<p>(f(x + h) - f(x)) / h</p> Signup and view all the answers

What is the Quadratic Formula?

<p>[-b ± √(b² - 4ac)] / (2a)</p> Signup and view all the answers

What is the Direct Variation formula?

<p>y = kx</p> Signup and view all the answers

What is the Inverse Variation formula?

<p>y = k / x</p> Signup and view all the answers

What does (f + g)(x) represent?

<p>f(x) + g(x)</p> Signup and view all the answers

What does (f - g)(x) represent?

<p>f(x) - g(x)</p> Signup and view all the answers

What does (fg)(x) represent?

<p>f(x) * g(x)</p> Signup and view all the answers

What does (f / g)(x) represent?

<p>f(x) / g(x)</p> Signup and view all the answers

What does (f º g)(x) represent?

<p>f(g(x))</p> Signup and view all the answers

What does y = -f(x) represent?

<p>Reflected across x-axis</p> Signup and view all the answers

What does y = f(-x) represent?

<p>Reflected across y-axis</p> Signup and view all the answers

What is an Even Function?

<p>f(-x) = f(x)</p> Signup and view all the answers

What is an Odd Function?

<p>f(-x) = -f(x)</p> Signup and view all the answers

What is Log Form?

<p>log₁₀ x = y</p> Signup and view all the answers

What is Exponent Form?

<p>10^y = x</p> Signup and view all the answers

What is the Change of Base Formula?

<p>log₅ M = log₁₀ M / log₁₀ 5</p> Signup and view all the answers

What is the Determinant of a 2x2 Matrix?

<p>| a d | = ad - bc</p> Signup and view all the answers

Flashcards

What is the Slope-Intercept Form of a Linear Equation?

The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope (steepness of the line) and b represents the y-intercept (where the line crosses the y-axis).

What is the Slope-Point Form of a Linear Equation?

The slope-point form of a linear equation is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope.

What is the Slope Formula?

The slope of a line between two points can be calculated using the slope formula: **m = (y₂ - y₁) / (x₂ - x₁) **. This formula measures the steepness of the line between those two points.

What is the relationship between the slopes of perpendicular lines?

If two lines are perpendicular, their slopes are negative reciprocals of each other. This means that if the slope of one line is m₁, the slope of the perpendicular line m₂ will be -1 / m₁.

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What is the Distance Formula?

Calculating the distance between two points follows the distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²] Applying this formula will provide the distance between the two points in a plane.

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What is the Midpoint Formula?

The midpoint formula is used to find the exact middle point between two given points. It is defined as (x₁ + x₂ / 2, y₁ + y₂ / 2). This formula will give you the coordinates of the midpoint.

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What is the Circle Formula?

The circle formula is represented as (x - h)² + (y - k)² = r² where (h, k) represents the coordinates of the center of the circle and r represents the radius.

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What is a complex number?

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1).

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What is the Difference Quotient?

The difference quotient is a critical expression used in calculus to find the derivative of a function. It is calculated as (f(x + h) - f(x)) / h, where f(x) is the function and h is a small change in x.

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What is the Quadratic Formula?

The quadratic formula is used to solve for the roots of a quadratic equation: ax² + bx + c = 0. The formula provides the solutions for x: x = [-b ± √(b² - 4ac)] / (2a).

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What is Direct Variation?

Direct variation occurs when two quantities are directly proportional, meaning that as one increases, the other increases by a constant factor. It is represented by the equation y = kx, where k is the constant of variation.

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What is Inverse Variation?

Inverse variation is a relationship between two quantities where as one increases, the other decreases by a factor that is inversely proportional. This is written as y = k / x, where k is the constant of variation.

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What is Function Addition?

Function addition involves combining two functions to create a new function. It is expressed as (f + g)(x) = f(x) + g(x), where f(x) and g(x) are the original functions.

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What is Function Subtraction?

Function subtraction involves determining the difference between two functions. It is represented by (f - g)(x) = f(x) - g(x), where f(x) and g(x) are the functions being subtracted.

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What is Function Multiplication?

Function multiplication involves finding the product of two functions, resulting in a new function. It is expressed as (fg)(x) = f(x) * g(x), where f(x) and g(x) are the original functions.

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What is Function Division?

Function division involves dividing one function by another, resulting in a new function. It is represented by (f / g)(x) = f(x) / g(x), where f(x) is the numerator and g(x) is the denominator.

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What is Function Composition?

Function composition involves combining two functions by substituting one function into another. It is expressed as (f º g)(x) = f(g(x)), where the output of g(x) becomes the input of f(x).

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What is a reflection across the x-axis in function transformations?

A reflection across the x-axis transforms a function by flipping it vertically. This is represented by y = -f(x), where the function f(x) is reflected over the x-axis.

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What is a reflection across the y-axis in function transformations?

A reflection across the y-axis flips a function horizontally. This is represented by y = f(-x), where the function f(x) is reflected across the y-axis.

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What is an even function?

Even functions are symmetric about the y-axis. This means that for any input x, the function's output is the same as for the negative input -x. This is expressed as f(-x) = f(x).

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What is an odd function?

Odd functions are symmetric about the origin. This means that the value of the function at -x is the opposite of the value at x. This is expressed as f(-x) = -f(x).

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What is the Logarithmic Form?

The logarithmic form log₁₀ x = y represents the base-10 logarithm of x, meaning 10 raised to the power of y equals x.

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What is the Exponent Form?

The exponent form 10^y = x is equivalent to the logarithmic form log₁₀ x = y. This form emphasizes that y is the exponent needed to raise the base 10 to get x.

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What is the Change of Base Formula?

The change of base formula log₅ M = log₁₀ M / log₁₀ 5 allows you to convert a logarithm from one base to another. This formula lets you calculate logarithms to different bases using base-10 logarithms.

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How do you calculate the determinant of a 2x2 Matrix?

For a 2x2 matrix, the determinant is calculated using the formula: |a d| |b c| = ad - bc. The determinant provides a scalar value that reveals important properties of the matrix.

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Study Notes

Linear Equations

  • Slope-Intercept Form: Expressed as y = mx + b, where m is the slope and b is the y-intercept.
  • Slope-Point Form: Given by y - y₁ = m(x - x₁), representing a linear equation using a point (x₁, y₁) and slope m.
  • Slope Formula: Calculated as (y₂ - y₁) / (x₂ - x₁), determines the steepness of a line between two points.

Line Relationships

  • Perpendicular Lines: If two lines L₁ and L₂ are perpendicular, their slopes satisfy m₁ = -1 / m₂.

Motion and Financial Calculations

  • Speed-Time-Distance: d = rt, where d is distance, r is rate, and t is time.
  • Simple Interest: Calculated as I = Prt, where I is interest, P is principal, r is rate, and t is time.
  • Compounded Interest: A = P(1 + r/n)^(nt) reflects interest added at specified intervals.

Exponential Decay

  • Carbon-14 Decay Rate: Approximately -0.00012, representing the rate of decay for carbon-14 isotopes.
  • Continuously Compounded Interest: Utilized with A = Pe^rt, where e is the base of natural logarithms.
  • Half-Life Formula: Expressed as 1/2 = e^rt, denotes the time required for half of a substance to decay.
  • Doubling Time Formula: Stated as 2 = e^rt, indicates the time required for a quantity to double.

Geometry and Distance

  • Distance Formula: Determined with d = √[(x₂ - x₁)² + (y₂ - y₁)²], representing the distance between two points in a plane.
  • Midpoint Formula: Given by (x₁ + x₂/2, y₁ + y₂/2), calculates the midpoint between two points.
  • Circle Formula: Represented as (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Algebraic Concepts

  • Complex Numbers: Formed as a + bi, combining real and imaginary parts.
  • Difference Quotient: Calculated as (f(x + h) - f(x)) / h, essential in finding derivatives.

Quadratic Equations

  • Quadratic Formula: Solving for x in ax² + bx + c = 0 provides x = [-b ± √(b² - 4ac)] / (2a).

Variation Relations

  • Direct Variation: Expressed as y = kx, indicating a proportional relationship where k is constant.
  • Inverse Variation: Written as y = k / x, showing a relationship where y decreases as x increases.

Function Operations

  • Function Addition: (f + g)(x) = f(x) + g(x), combining two functions.
  • Function Subtraction: (f - g)(x) = f(x) - g(x), determining the difference between two functions.
  • Function Multiplication: (fg)(x) = f(x) * g(x), represents the product of two functions.
  • Function Division: (f / g)(x) = f(x) / g(x), portraying the division of two functions.
  • Function Composition: (f º g)(x) = f(g(x)), combining two functions by substituting.

Function Transformations

  • Reflection Across X-axis: y = -f(x) indicates a vertical reflection of the function.
  • Reflection Across Y-axis: y = f(-x) indicates a horizontal reflection of the function.

Function Symmetry

  • Even Function: Characterized by f(-x) = f(x), symmetrical around the y-axis.
  • Odd Function: Identified by f(-x) = -f(x), symmetrical around the origin.

Logarithmic Functions

  • Logarithmic Form: log₁₀ x = y denotes the logarithm of x to the base 10.
  • Exponent Form: 10^y = x transforms the logarithmic equation to exponential.
  • Change of Base Formula: log₅ M = log₁₀ M / log₁₀ 5 allows conversion between logarithmic bases.

Matrix Calculations

  • Determinant of 2x2 Matrix: Calculated using | a d | | b c | = ad - bc, providing a scalar value representing matrix properties.

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Test your knowledge of essential precalculus formulas with these flashcards. Covering concepts such as slope-intercept form, slope-point form, and various interest calculations, these cards are a great way to reinforce your understanding. Perfect for students looking to master precalculus concepts quickly and effectively.

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