Podcast
Questions and Answers
Which statement accurately describes a linear function?
Which statement accurately describes a linear function?
- It forms a curved line on a graph.
- It is a trigonometric function.
- It is a quadratic function with degree 2.
- It forms a straight line on a graph. (correct)
A linear function can only represent relationships where the dependent variable increases as the independent variable increases.
A linear function can only represent relationships where the dependent variable increases as the independent variable increases.
False (B)
What is the degree of a polynomial function that determines a linear function?
What is the degree of a polynomial function that determines a linear function?
1 or 0
In the slope-intercept form of a linear equation, y = mx + b, the variable m represents the ______ of the line.
In the slope-intercept form of a linear equation, y = mx + b, the variable m represents the ______ of the line.
Match the following forms of linear equations with their descriptions:
Match the following forms of linear equations with their descriptions:
Given two points on a line, (2, 5) and (6, 3), what is the slope of the line?
Given two points on a line, (2, 5) and (6, 3), what is the slope of the line?
If a line has a positive slope, it will always increase from left to right on a graph.
If a line has a positive slope, it will always increase from left to right on a graph.
What does 'b' represent in the slope-intercept form of a linear equation, y = mx + b?
What does 'b' represent in the slope-intercept form of a linear equation, y = mx + b?
The formula to calculate the slope (m) of a line given two points ($x_1$, $y_1$) and ($x_2$, $y_2$) is m = ($y_2$ - $y_1$) / (_______).
The formula to calculate the slope (m) of a line given two points ($x_1$, $y_1$) and ($x_2$, $y_2$) is m = ($y_2$ - $y_1$) / (_______).
What is the y-intercept of the linear function defined by f(x) = -(1/2)x + 6?
What is the y-intercept of the linear function defined by f(x) = -(1/2)x + 6?
Graphs cannot be used to represent real-world situations.
Graphs cannot be used to represent real-world situations.
In the context of interpreting graphs, what does the 'rate' refer to?
In the context of interpreting graphs, what does the 'rate' refer to?
A straight-line graph is a ________ representation of a linear function.
A straight-line graph is a ________ representation of a linear function.
Given the general equation of a straight line as y = mx + c, what do 'm' and 'c' represent, respectively?
Given the general equation of a straight line as y = mx + c, what do 'm' and 'c' represent, respectively?
Match the description to the correct formula.
Match the description to the correct formula.
Flashcards
Linear Function
Linear Function
A function that forms a straight line when graphed. It's generally a polynomial function with a degree of at most 1 or 0.
Standard Form
Standard Form
A way to express a linear equation using integers A, B, and C
Slope-Intercept Form
Slope-Intercept Form
Shows the slope (m) and y-intercept (b) of a linear equation.
Point-Slope Form
Point-Slope Form
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Straight-line graph
Straight-line graph
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Midpoint
Midpoint
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Interpreting Graphs
Interpreting Graphs
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Study Notes
- The focus is on graphing of linear functions.
- The learning objectives include plotting tables of values and graphs of linear functions, and representing real-world situations using graphs of linear functions.
Linear Functions
- These are functions that form a straight line on a graph.
- They are generally polynomial functions.
- The degree is utmost 1 or 0.
Forms
- Forms include standard form, slope-intercept form, and point-slope form.
- Standard form is Ax + By = C, where A, B, and C are integers.
- Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
- Point-slope form is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point.
Example Problem
- Find an equation of the linear function given f(2) = 5 and f(6) = 3.
- First list the ordered pairs: (2, 5) and (6, 3).
- The slope is calculated as (3-5) / (6-2) = -2 / 4 = -1/2.
- Substitute the slope into the equation y = mx + c.
- 5 = -(1/2) (2) + b
- 5 = -1 + b
- b = 6, which is a y-intercept.
- Hence, y = -(1/2) (x) + 6
- In function notation: f(x) = -(1/2) (x) + 6
Graph in Practical Situation
- The learning objective is to interpret graphs related to rate and change.
- A straight-line graph is a visual representation of a linear function with a general equation of y = mx + c, where m is the gradient and c is the y-intercept.
- Example: y = 2x + 1 with x = (-2, -1, 0)
Midpoint of a Line Segment
- Midpoint of a line segment is a point exactly halfway between two points and equidistant from each endpoint of the straight-line segment.
- The midpoint is found by averaging the x-coordinates and the y-coordinates of the endpoints.
- For points (4, 6) and (8, 2), the midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).
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