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Questions and Answers
What is the slope and y-intercept of the function represented by the equation $f(x) = -3x + 5$?
What is the slope and y-intercept of the function represented by the equation $f(x) = -3x + 5$?
- Slope: -5, Y-intercept: 3
- Slope: -3, Y-intercept: 5 (correct)
- Slope: 3, Y-intercept: -5
- Slope: 5, Y-intercept: -3
Which of the following equations represents a quadratic function that opens downwards?
Which of the following equations represents a quadratic function that opens downwards?
- $f(x) = 3x^2 + 2x$
- $f(x) = 2x^2 - 5x + 4$
- $f(x) = 4x^2 - 2x + 1$
- $f(x) = -x^2 + 3$ (correct)
Determine the vertex of the quadratic function $f(x) = 2x^2 + 4x + 1$.
Determine the vertex of the quadratic function $f(x) = 2x^2 + 4x + 1$.
- Vertex: (1, 1)
- Vertex: (-1, 3)
- Vertex: (-1, -1) (correct)
- Vertex: (1, 3)
What can be inferred about the graph of the linear function $f(x) = 0.5x + 3$?
What can be inferred about the graph of the linear function $f(x) = 0.5x + 3$?
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In the quadratic function $f(x) = -3x^2 + 6x + 2$, what is the y-intercept?
In the quadratic function $f(x) = -3x^2 + 6x + 2$, what is the y-intercept?
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For the quadratic function $f(x) = x^2 + 6x + 8$, what is the axis of symmetry?
For the quadratic function $f(x) = x^2 + 6x + 8$, what is the axis of symmetry?
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Which characteristic is true for the linear equation $f(x) = -5$?
Which characteristic is true for the linear equation $f(x) = -5$?
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Which of the following functions has a positive slope?
Which of the following functions has a positive slope?
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Study Notes
Mathematics Functions
Linear Functions
- Definition: A function that graphs to a straight line.
- Standard Form:
- ( f(x) = mx + b )
- ( m ): slope (rate of change)
- ( b ): y-intercept (where the line crosses the y-axis)
- ( f(x) = mx + b )
- Characteristics:
- Slope indicates the direction of the line:
- Positive slope: line rises from left to right.
- Negative slope: line falls from left to right.
- Zero slope: horizontal line (no change).
- Graph: Straight line; can be increasing or decreasing.
- Slope indicates the direction of the line:
- Examples:
- ( f(x) = 2x + 3 ) (slope = 2, y-intercept = 3)
- ( f(x) = -x + 1 ) (slope = -1, y-intercept = 1)
Quadratic Functions
- Definition: A function that graphs to a parabola (U-shaped curve).
- Standard Form:
- ( f(x) = ax^2 + bx + c )
- ( a ): coefficient of ( x^2 ) (determines direction of the parabola)
- ( b ): coefficient of ( x )
- ( c ): constant term (y-intercept)
- ( f(x) = ax^2 + bx + c )
- Characteristics:
- Opens upwards if ( a > 0 ) and downwards if ( a < 0 ).
- Vertex: The highest or lowest point of the parabola, calculated using ( x = -\frac{b}{2a} ).
- Axis of symmetry: Vertical line ( x = -\frac{b}{2a} ).
- Y-intercept: Found by evaluating ( f(0) = c ).
- Examples:
- ( f(x) = x^2 - 4x + 3 ) (opens upwards, vertex at ( (2, -1) ))
- ( f(x) = -2x^2 + 4x - 1 ) (opens downwards, vertex at ( (1, 1) ))
Linear Functions
- Definition: Represents a relationship where a variable changes at a constant rate, producing a straight line on a graph.
- Standard Form:
- ( f(x) = mx + b )
- ( m ): Represents the slope, indicating how steep the line is.
- ( b ): The y-intercept, the point at which the line intersects the y-axis.
- ( f(x) = mx + b )
- Slope Characteristics:
- Positive slope indicates the line ascends from left to right.
- Negative slope indicates the line descends from left to right.
- Zero slope indicates a horizontal line, signifying no change in the y-value with different x-values.
- Graph Properties: Consists of straight lines that can be either increasing or decreasing based on the slope.
- Examples:
- For ( f(x) = 2x + 3 ): slope is 2 (line rises) and y-intercept is 3 (crosses y-axis at (0,3)).
- For ( f(x) = -x + 1 ): slope is -1 (line falls) and y-intercept is 1 (crosses y-axis at (0,1)).
Quadratic Functions
- Definition: Describe a parabolic relationship, yielding a U-shaped graph.
- Standard Form:
- ( f(x) = ax^2 + bx + c )
- ( a ): Determines whether the parabola opens upwards (if ( a > 0 )) or downwards (if ( a < 0 )).
- ( b ): Influences the position of the vertex.
- ( c ): The y-intercept, where the parabola crosses the y-axis.
- ( f(x) = ax^2 + bx + c )
- Vertex: The highest or lowest point on the parabola, calculated using ( x = -\frac{b}{2a} ).
- Axis of Symmetry: A vertical line represented by ( x = -\frac{b}{2a} ) that divides the parabola into two mirror-image halves.
- Y-intercept: Detected by evaluating the function at zero, ( f(0) = c ).
- Examples:
- For ( f(x) = x^2 - 4x + 3 ): opens upwards and has a vertex at ( (2, -1) ).
- For ( f(x) = -2x^2 + 4x - 1 ): opens downwards and has a vertex at ( (1, 1) ).
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