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Questions and Answers
What is the quadratic function equation?
What is the quadratic function equation?
f(x) = ax^2 + bx + c
The steps of graphing a quadratic equation include finding the y-intercept (0, _____).
The steps of graphing a quadratic equation include finding the y-intercept (0, _____).
c
What do you need to do to find the roots of a quadratic equation?
What do you need to do to find the roots of a quadratic equation?
- Graph the function
- Find the zeros of the related quadratic function
- Factor the equation
- All of the above (correct)
The sum of squares is factorable.
The sum of squares is factorable.
What is a double root?
What is a double root?
What does i^2 equal?
What does i^2 equal?
How should complex numbers be written?
How should complex numbers be written?
If the value of the discriminant is zero, how many real roots are there?
If the value of the discriminant is zero, how many real roots are there?
If the value of the discriminant is negative, how many roots are there?
If the value of the discriminant is negative, how many roots are there?
If the value of the discriminant is a positive perfect square, how many roots are there?
If the value of the discriminant is a positive perfect square, how many roots are there?
What steps are involved in solving quadratic inequalities algebraically?
What steps are involved in solving quadratic inequalities algebraically?
The term 'less than' refers to values that are _____ (below).
The term 'less than' refers to values that are _____ (below).
The term 'greater than' indicates values that are _____ (above).
The term 'greater than' indicates values that are _____ (above).
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Study Notes
Quadratic Function Equation
- Defined as f(x) = ax² + bx + c, where a, b, and c are constants.
Steps of Graphing a Quadratic Equation
- Y-Intercept: Calculate at (0, c).
- Axis of Symmetry: Found using x = -b / (2a).
- Vertex: Located at (aos, f(aos)), where aos is the axis of symmetry.
- Table of Values: Create a table to aid in plotting the graph.
- Graph the Function: Plot the points and sketch the parabola.
- Domain and Range: Domain includes all real numbers; range varies based on the vertex's location.
Roots
- Solutions of a quadratic equation found by determining the ZEROS of the related quadratic function.
- Zeros are the x-intercepts of the graph, identified by where y = 0.
Sum of Squares
- Not factorable in the context of quadratic equations.
Double Root
- Occurs when a quadratic equation has only one unique solution, indicated by the graph touching the x-axis at one point.
Pure Imaginary Numbers
- Defined by i² = -1; crucial for understanding complex numbers.
Remainder Pattern
- Recognizes the cyclical nature when dividing by polynomials involving i: i, -1, -i, 1.
Adding and Subtracting Complex Numbers
- Expressed in the form a + bi, treating i as a variable; a and b represent real numbers.
Discriminant Values
- Zero: Indicates 1 real and rational root; the graph touches the x-axis at one point.
- Negative: Results in 2 complex and irrational roots; the graph does not touch the x-axis.
- Positive Perfect Square: Results in 2 real and rational roots.
- Positive but Not Perfect Square: Indicates 2 real and irrational roots.
Solving Quadratic Inequalities Algebraically
- Find roots using factoring, completing the square, or the quadratic formula.
- Roots divide the number line into 3 intervals.
- Test a value within each interval to check for solutions.
- If ≥ or ≤ applies, roots are included in the solution set.
Less Than
- Represents values that fall below a certain point on a number line.
Greater Than
- Represents values that exceed a certain point on a number line.
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