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Questions and Answers
Consider a system of two linear equations. Which of the following scenarios indicates that the system has infinitely many solutions?
Consider a system of two linear equations. Which of the following scenarios indicates that the system has infinitely many solutions?
- The slopes of the two lines are negative reciprocals, indicating perpendicular lines.
- The two equations are scalar multiples of each other, representing the same line. (correct)
- The substitution method leads to a quadratic equation with two distinct real roots.
- The elimination method results in an equation of the form $0 = c$, where $c$ is a non-zero constant.
A line is defined by the equation $y = mx + b$. If the line is shifted vertically upwards by $k$ units and reflected over the x-axis, what is the equation of the new line?
A line is defined by the equation $y = mx + b$. If the line is shifted vertically upwards by $k$ units and reflected over the x-axis, what is the equation of the new line?
- $y = -mx - b - k$ (correct)
- $y = mx + b + k$
- $y = -mx + b - k$
- $y = -mx - b + k$
Given two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line, how does changing the order of subtraction in the slope formula affect the result?
Given two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line, how does changing the order of subtraction in the slope formula affect the result?
- The slope remains unchanged.
- The value of the slope becomes the reciprocal of the original slope.
- The sign of the slope changes, but the absolute value remains the same. (correct)
- The slope becomes undefined.
Which of the following equations represents a line that is parallel to $2x + 3y = 6$ and passes through the point $(0, 0)$?
Which of the following equations represents a line that is parallel to $2x + 3y = 6$ and passes through the point $(0, 0)$?
A relation is defined by the set of ordered pairs ${(1, 2), (2, 3), (3, 4), (1, 5)}$. Why does this relation fail to be a function?
A relation is defined by the set of ordered pairs ${(1, 2), (2, 3), (3, 4), (1, 5)}$. Why does this relation fail to be a function?
If $f(x) = x^2 - 3x + 2$, what is the value of $f(a + 1)$?
If $f(x) = x^2 - 3x + 2$, what is the value of $f(a + 1)$?
The graph of $y = f(x)$ is transformed to $y = f(x + 2) - 3$. Describe the transformation in words.
The graph of $y = f(x)$ is transformed to $y = f(x + 2) - 3$. Describe the transformation in words.
A graph of a function $y = f(x)$ is reflected over the y-axis. How does this transformation affect the coordinates of a point $(a, b)$ on the original graph?
A graph of a function $y = f(x)$ is reflected over the y-axis. How does this transformation affect the coordinates of a point $(a, b)$ on the original graph?
The function $g(x) = 3f(x)$ represents a transformation of the graph of $y = f(x)$. Describe the transformation.
The function $g(x) = 3f(x)$ represents a transformation of the graph of $y = f(x)$. Describe the transformation.
Which of the following transformations will result in the graph of $y = x^2$ being wider than the original graph?
Which of the following transformations will result in the graph of $y = x^2$ being wider than the original graph?
Determine whether the function $f(x) = x^4 + 2x^2 + 1$ is even, odd, or neither.
Determine whether the function $f(x) = x^4 + 2x^2 + 1$ is even, odd, or neither.
Which of the following functions is odd?
Which of the following functions is odd?
A piecewise function is defined as follows: $f(x) = \begin{cases} x^2, & \text{if } x < 0 \ 2x + 1, & \text{if } x \geq 0 \end{cases}$. What is the value of $f(-2) + f(3)$?
A piecewise function is defined as follows: $f(x) = \begin{cases} x^2, & \text{if } x < 0 \ 2x + 1, & \text{if } x \geq 0 \end{cases}$. What is the value of $f(-2) + f(3)$?
A line has a slope of $m = -2$ and passes through the point $(1, 4)$. What is the equation of this line in standard form?
A line has a slope of $m = -2$ and passes through the point $(1, 4)$. What is the equation of this line in standard form?
Which of the following represents a vertical compression of the function $f(x) = |x|$ by a factor of $\frac{1}{2}$?
Which of the following represents a vertical compression of the function $f(x) = |x|$ by a factor of $\frac{1}{2}$?
Given the function $f(x) = x^3 - ax$, find the value(s) of $a$ for which the function is odd.
Given the function $f(x) = x^3 - ax$, find the value(s) of $a$ for which the function is odd.
A line $L_1$ has the equation $y = 3x - 5$. What is the equation of a line $L_2$ that is perpendicular to $L_1$ and passes through the point $(3,1)$?
A line $L_1$ has the equation $y = 3x - 5$. What is the equation of a line $L_2$ that is perpendicular to $L_1$ and passes through the point $(3,1)$?
A graph of $y = f(x)$ is stretched horizontally by a factor of 2. What is the equation of the transformed graph?
A graph of $y = f(x)$ is stretched horizontally by a factor of 2. What is the equation of the transformed graph?
Consider the piecewise function $f(x) = \begin{cases} x + 1, & \text{if } x \leq 1 \ x^2, & \text{if } x > 1 \end{cases}$. Is this function continuous at $x = 1$?
Consider the piecewise function $f(x) = \begin{cases} x + 1, & \text{if } x \leq 1 \ x^2, & \text{if } x > 1 \end{cases}$. Is this function continuous at $x = 1$?
The domain of a function represents:
The domain of a function represents:
Flashcards
Linear Equation
Linear Equation
A linear equation in two variables is written as ax + by = c.
Substitution Method
Substitution Method
Solve for one variable in terms of the other, then substitute into the second equation.
Elimination Method
Elimination Method
Multiply equations to align coefficients, then add or subtract to eliminate one variable.
Slope-Intercept Form
Slope-Intercept Form
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Point-Slope Form
Point-Slope Form
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Standard Form (Line)
Standard Form (Line)
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Slope Formula
Slope Formula
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Parallel Lines
Parallel Lines
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Perpendicular Lines
Perpendicular Lines
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Function
Function
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Vertical Line Test
Vertical Line Test
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Function Notation f(c)
Function Notation f(c)
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Domain
Domain
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Range
Range
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Vertical Shift
Vertical Shift
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Horizontal Shift
Horizontal Shift
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Reflection over x-axis
Reflection over x-axis
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Reflection over y-axis
Reflection over y-axis
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Even Function
Even Function
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Odd Function
Odd Function
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Study Notes
- Key formulas and methods for solving problems related to linear equations, writing equations of lines, functions and relations, and transformations of graphs.
Linear Equations in Two Variables
- A linear equation in two variables takes the form (ax + by = c).
Methods to Solve Systems of Equations
Substitution Method
- Solve one equation for one variable in terms of the other variable, resulting in an equation such as (y = mx + b).
- Substitute the expression into the second equation.
- Solve the resulting equation for the remaining variable.
- Substitute the value back into either original equation to find the value of the other variable.
Elimination Method
- Multiply one or both equations by constants to make the coefficients of one variable match or be opposites.
- Add or subtract the equations to eliminate one variable.
- Solve for the remaining variable.
- Substitute the value back into either original equation to find the value of the eliminated variable.
Writing the Equation of a Line
Slope-Intercept Form
- The equation is (y = mx + b).
- (m) represents the slope.
- (b) represents the y-intercept.
Point-Slope Form
- Used when given a point ((x_1, y_1)) and slope (m).
- The equation is (y - y_1 = m(x - x_1)).
Standard Form
- The equation is (Ax + By = C).
- (A), (B), and (C) are integers.
Slope Formula
- Used when given two points ((x_1, y_1)) and ((x_2, y_2)).
- The equation is (m = \frac{y_2 - y_1}{x_2 - x_1}).
Parallel & Perpendicular Lines
- Parallel lines have the same slope: (m_1 = m_2).
- Perpendicular lines have opposite reciprocal slopes, such that (m_1 \times m_2 = -1).
Functions and Relations
- A function is a relation where each input (x-value) has exactly one output (y-value).
Vertical Line Test
- A relation is a function if no vertical line intersects the graph more than once.
Function Notation
- If (f(x) = ax + b), then (f(c)) means substituting (x = c) into the equation.
Domain & Range
- The domain includes all possible (x)-values.
- The range includes all possible (y)-values.
Transformations of Graphs
Vertical and Horizontal Shifts
- For a vertical shift, (y = f(x) + c); (+c) moves the graph up, and (-c) moves it down.
- For a horizontal shift, (y = f(x - c)); (-c) moves the graph right, and (+c) moves it left.
Reflection
- Over the x-axis, (y = -f(x)).
- Over the y-axis, (y = f(-x)).
Vertical Stretch/Compression
- (y = a f(x)) represents a vertical stretch or compression.
- If (|a| > 1), the graph stretches vertically.
- If (0 < |a| < 1), the graph compresses vertically.
Horizontal Stretch/Compression
- (y = f(bx)) represents a horizontal stretch or compression.
- If (|b| > 1), the graph shrinks horizontally.
- If (0 < |b| < 1), the graph stretches horizontally.
Symmetry, Even/Odd, and Piecewise-Defined Functions
Even Functions
- Symmetric about the y-axis.
- Defined by (f(-x) = f(x)).
Odd Functions
- Symmetric about the origin.
- Defined by (f(-x) = -f(x)).
Piecewise-Defined Function
- A function defined by different equations for different intervals of its domain: [ f(x) = \begin{cases} \text{Equation 1}, & \text{if } x \text{ in range 1} \ \text{Equation 2}, & \text{if } x \text{ in range 2} \end{cases} ]
Solving Linear Equations:
- Identify the equations in the form (ax + by = c).
- Use either the substitution or elimination method.
- Solve for the remaining variable.
- Substitute back to find the second variable.
- Write the solution as an ordered pair ((x, y)).
Tip
- No solution exists if the equations result in a contradiction.
- Infinite solutions exists if the equations are identical.
Writing Equations of Lines
- Find the slope (m = \frac{y_2 - y_1}{x_2 - x_1}) using two given points.
- Use the point-slope form (y - y_1 = m(x - x_1)).
- Simplify into slope-intercept form (y = mx + b), where (b) is the y-intercept.
Tip
- Plug directly into point-slope form if given the slope and a point.
Functions and Relations
- A relation is a set of ordered pairs, while a function requires each input (x) to have only one output (y).
- To check if a relation is a function:
- It is not a function if any (x) value repeats with different (y) values.
- Use the vertical line test on a graph; it is not a function if a vertical line touches the graph more than once.
Tip
- (f(x)) replaces (y), for example, (y = 2x + 3) is the same as (f(x) = 2x + 3).
Transformations of Graphs
Types of Transformations:
-
Translation (Shift):
- Up/Down: (y = f(x) + k), where (k > 0) shifts up and (k < 0) shifts down.
- Left/Right: (y = f(x - h)), where (h > 0) shifts right and (h < 0) shifts left.
-
Reflection:
- Over the x-axis: (y = -f(x)).
- Over the y-axis: (y = f(-x)).
-
Stretch/Shrink:
- Vertical Stretch if (a > 1): (y = a f(x)).
- Vertical Shrink if (0 < a < 1): (y = a f(x)).
Tip
- Analyze the base graph before applying transformations.
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