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Questions and Answers
What is the vertex of the parabola if it is represented by the vertex form equation?
What is the vertex of the parabola if it is represented by the vertex form equation?
(h, k) where h is the x-coordinate and k is the y-coordinate of the vertex.
What defines the axis of symmetry for a quadratic function?
What defines the axis of symmetry for a quadratic function?
The axis of symmetry is the vertical line x = h, where (h, k) is the vertex.
How would you express the quadratic function in factored form?
How would you express the quadratic function in factored form?
The factored form is expressed as y = a(x - r1)(x - r2), where r1 and r2 are the roots.
What is the significance of the roots or solutions of a quadratic function?
What is the significance of the roots or solutions of a quadratic function?
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If the equation x^2 + 8x + 9 is solved by completing the square, what does it transform into?
If the equation x^2 + 8x + 9 is solved by completing the square, what does it transform into?
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What degree classification does the polynomial –2x^4 – x^3 + 8x^2 + 12 have?
What degree classification does the polynomial –2x^4 – x^3 + 8x^2 + 12 have?
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How can you identify the number of terms in the polynomial 8x^4 + 7x^3 + 5x^2 + 8?
How can you identify the number of terms in the polynomial 8x^4 + 7x^3 + 5x^2 + 8?
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What does the imaginary number i represent in mathematics?
What does the imaginary number i represent in mathematics?
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What is the effect of a negative sign in front of a quadratic function on its graph?
What is the effect of a negative sign in front of a quadratic function on its graph?
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How do you determine the axis of symmetry for a quadratic function in standard form?
How do you determine the axis of symmetry for a quadratic function in standard form?
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In the function transformation $f(x) = a(x-h)^2 + k$, what do the variables h and k represent?
In the function transformation $f(x) = a(x-h)^2 + k$, what do the variables h and k represent?
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What is the significance of the vertex in a parabola?
What is the significance of the vertex in a parabola?
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When simplifying the expression $\frac{x^2 - 4}{x - 2}$, what restrictions apply to x?
When simplifying the expression $\frac{x^2 - 4}{x - 2}$, what restrictions apply to x?
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If the range of a function is all real numbers except 6, what can you infer about its maximum or minimum value?
If the range of a function is all real numbers except 6, what can you infer about its maximum or minimum value?
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How does the presence of a factor $(x-4)$ in a rational function affect its vertical asymptote?
How does the presence of a factor $(x-4)$ in a rational function affect its vertical asymptote?
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What does it indicate if a quadratic function has an axis of symmetry at x = 9?
What does it indicate if a quadratic function has an axis of symmetry at x = 9?
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Is the polynomial $3x^2 - 3x + 3$ a polynomial function? If yes, state its degree, type, and leading coefficient.
Is the polynomial $3x^2 - 3x + 3$ a polynomial function? If yes, state its degree, type, and leading coefficient.
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Determine the domain and range of the function $f(x) = \frac{1}{x}$.
Determine the domain and range of the function $f(x) = \frac{1}{x}$.
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What is the leading coefficient and degree of the polynomial $2x^2 + 3x - 5$?
What is the leading coefficient and degree of the polynomial $2x^2 + 3x - 5$?
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Write the polynomial function of least degree that has the zeros at -3 and 2.
Write the polynomial function of least degree that has the zeros at -3 and 2.
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Identify which of the following functions has a slant asymptote: $f(x) = \frac{2x + 1}{x - 3}$ or $g(x) = \frac{x^2 - 2x - 3}{x + 3}$.
Identify which of the following functions has a slant asymptote: $f(x) = \frac{2x + 1}{x - 3}$ or $g(x) = \frac{x^2 - 2x - 3}{x + 3}$.
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What is the product $f(x) \cdot g(x)$ if $f(x) = 3x^2 - 3x + 3$ and $g(x) = 2x + 5$?
What is the product $f(x) \cdot g(x)$ if $f(x) = 3x^2 - 3x + 3$ and $g(x) = 2x + 5$?
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Using synthetic division, what is the remainder when dividing $2x^3 - 5x^2 - 4x - 25$ by $x - 4$?
Using synthetic division, what is the remainder when dividing $2x^3 - 5x^2 - 4x - 25$ by $x - 4$?
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If the graph of a function appears to have an absolute minimum and is symmetric about the y-axis, what type of function could it represent?
If the graph of a function appears to have an absolute minimum and is symmetric about the y-axis, what type of function could it represent?
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Study Notes
Algebra 2 - Semester 1 Final A
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Y-intercept: An ordered pair where the graph crosses the y-axis. The x-coordinate is always 0.
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Domain: The set of all x-coordinates in a relation or function.
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Range: The set of all y-coordinates in a relation or function.
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Example Relation Range: For the relation {(-4, 1), (-2, 0), (8, -1)}, the range is {-1, 0, 1}.
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Example Relation Domain: For the relation shown in question 5, the domain is {-2, -1, 0, 1, 2, 3}.
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Graphing Relations: Plotting ordered pairs on a graph to understand relationships between x and y values.
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Vertical Line Test: Used to determine if a graph represents a function. If any vertical line intersects the graph in more than one point, it is not a function.
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Translations (Graph Transformations): Shifting a graph horizontally or vertically. A shift to the left adds to the x-value, a shift to the right subtracts from the x-value, a shift up adds to the y-value, and a shift down subtracts from the y-value.
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Examples of Function Transformations
- (x-4): shift right 4 units
- (x+4): shift left 4 units
- f(x) - 4: shift down 4 units
- f(x) + 4: shift up 4 units.
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Vertex form of a quadratic function: y = a(x-h)^2 + k, where (h, k) is the vertex.
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Factored form: An expression written as a product of polynomials.
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Example of finding the vertex and axis of symmetry: The vertex of y= 2(x+2)^2-4 is (-2,-4), and the axis of symmetry is x=-2.
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Quadratic Function Solutions (Roots/Zeros): Solutions to quadratic equations, often found using the quadratic formula or factoring.
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Quadratic Formula: A formula that gives the solutions to quadratic equations in the form ax² + bx + c = 0.
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Imaginary Numbers (i): i = √-1
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Classifying Polynomials (by degree):
- Linear: degree 1
- Quadratic: degree 2
- Cubic: degree 3
- Quartic: degree 4
- Quintic: degree 5
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Synthetic Division: A method for dividing polynomials by a linear factor.
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Graphing polynomial functions: Determining the zeros and shape.
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Important concepts from the last questions: Finding the graph that represents the polynomial, factoring to find the appropriate values, and writing polynomial equations.
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Description
Test your understanding of key concepts in Algebra 2, including y-intercept, domain, range, and graph transformations. This quiz covers vital topics such as the Vertical Line Test and essential graphing techniques crucial for mastering relations and functions.
