Questions and Answers
How do you factor a difference of squares?
(x + 6)(x - 6)
What is the equation of a parabola in vertex form?
y = a(x - h)² + k
How do you find the x-point of the vertex of a parabola when you have ax² + bx + c?
x = -b ÷ 2a
You can tell if there is a double root of a function if the graph just touches the x-axis.
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When a line goes through the x-axis on a graph, it indicates a regular root.
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What does 'the value of a function' refer to?
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What is the max value of a function?
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What does it mean when a function has 4 'roots'?
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What do you do whenever you have i²?
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How would you solve (x + 7)² = 25?
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What is the formula for exponential growth?
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What is the standard form of a trinomial?
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What must be true for a quadratic equation to have only one solution?
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What must be true for a quadratic equation to have two solutions?
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What must be true for a quadratic equation to have no solutions?
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When solving a quadratic equation, only the positive root is needed when dealing with a constant.
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How do you solve quadratic equations?
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Quadratic equations have x² while linear equations do not.
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What are the steps to find the minimum or maximum value of a parabola?
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√(3y) can be simplified to √3 √y.
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What is x^(1/2)?
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How do you determine the vertex of a parabola from its equation?
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What is b^(3/4)?
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How does a function g(x) look when it is factored?
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How do you identify a factor of g(x)?
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Adding $50 per week indicates linear growth.
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Adding 5% of your current value each year indicates exponential growth.
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Gaining 5% of your initial value each time means linear growth.
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What is the percent increase if the price of bread is multiplied by 1.9?
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How do you make an expression undefined?
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Study Notes
Factoring and Quadratics
- Factor a difference of squares using the formula: ( (a+b)(a-b) ), e.g., ( x² - 36 = (x+6)(x-6) ).
- The vertex form of a parabola is given by ( y = a(x - h)² + k ), where ( (h, k) ) is the vertex.
- The x-coordinate of the vertex can be found using the formula ( x = -b/(2a) ).
Roots and Values
- A "double root" occurs when the graph touches the x-axis without crossing it.
- A function crosses the x-axis for every regular root present.
- The value of a function refers to its y-value; maximum value is indicated by the largest y-value.
Roots and Factors
- A function with four roots means it intersects or touches the x-axis four times.
- For any ( i^2 ), replace it with -1.
- To solve squared variable equations, take the square root of both sides, leading to two separate equations.
Methods and Determinants
- Factor by grouping involves organizing terms and finding common factors. E.g., from ( (x³ - 7x²) + (3x - 21) = 0 ), factor to ( (x-7)(x² + 3) = 0 ).
- The formula for exponential growth is ( A = Pe^{rt} ).
Identifying Quadratics
- In a quadratic equation ( ax² + bx + c = 0 ), ( a, b, c ) can be identified from the terms in the equation. For example, in ( 3x² - 4x + 5 ), ( a = 3 ), ( b = -4 ), ( c = 5 ).
- The standard form of a trinomial is ( ax² + bx + c ).
- The quadratic formula's discriminant ( b² - 4ac ) determines the nature of the roots:
- Equal to 0 gives one solution (double root).
- Positive gives two solutions (two roots).
- Negative gives zero solutions.
Solving Techniques
- To solve for maximum or minimum values of a parabola, graph the x-intercepts, average them for the vertex x-value, and substitute back to find the y-value.
- Understanding root multiplication: ( \sqrt{3y} = \sqrt{3}\sqrt{y} ) helps in distributed calculations.
Roots and Functions
- The vertex of a parabola in standard form can be solved by identifying the x-value from the transformation inside the function and the y-value from the constant outside.
- The expression ( b^{3/4} ) represents the 4th root of ( b ) cubed.
Growth Models
- Linear growth is exemplified by adding a constant amount (e.g., $50 each week).
- Exponential growth occurs when a percentage of the current value is added annually (e.g., 5% of the current value).
Miscellaneous
- A 90% increase results from multiplying a price by 1.9.
- An expression becomes undefined when the denominator equals zero; solve by setting the denominator to zero.
Studying That Suits You
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Description
Test your knowledge of Algebra 2 concepts with these SAT Math essentials flashcards. Focus on important topics such as factoring differences of squares and understanding the vertex form of parabolas. Perfect for quick revisions and exam preparation.
