11 Questions
What is the significance of the $y$intercept in the context of graphing linear equations?
It is the point at which the graph crosses the $y$axis.
Which form of a line's equation uses a specific point on the line and the slope to represent the line?
Pointslope form
How is the slope of a line defined?
The ratio of vertical change to horizontal change
Which aspect of a linear equation does the slopeintercept form emphasize?
The slope and $y$intercept
When finding equations of lines, what does the slope represent?
Vertical change to horizontal change ratio
What transformation is represented by the function f(x) = g(x3)?
Horizontal translation
Which transformation is indicated by the function f(x) = g(x)?
Vertical reflection
If f(x) = g(2x), what type of scaling is applied to the function?
Horizontal compression
What does the xintercept of a graph represent?
The point at which x = 0
How is a vertical shift represented in a function?
f(x) = g(x+k)
Which operation would result in a reflection over the yaxis in a function?
f(x) = g(x)
Study Notes
Transformations of Functions and Straight Line Graphs
Transformations of functions and their corresponding straight line graphs are fundamental concepts in understanding algebraic relationships. By manipulating functions through translations, reflections, and scalings, we can create a rich variety of new graphs to explore and analyze.
Transformations of Functions

Translation: A vertical shift moves the graph up or down by a fixed amount. A horizontal shift moves the graph left or right by a fixed amount. These shifts are represented by (f(x) = g(xh)) for a horizontal shift by (h) and (f(x) = g(x+k)) for a vertical shift by (k).

Reflection: Reflecting a function over the (x)axis or (y)axis changes the graph's orientation. A reflection over the (x)axis is indicated by changing the sign of the function's output ((f(x) = g(x))). A reflection over the (y)axis is indicated by changing the sign of the function's input ((f(x) = g(x))).

Scaling: A function can be stretched or compressed horizontally or vertically. A horizontal scaling by a factor of (a) is indicated by ((f(x) = g(ax))). A vertical scaling by a factor of (b) is indicated by ((f(x) = bg(x))).
Interpreting Graphs
To understand the geometric relationships between transformed graphs, we analyze the intercepts and the slopes of the lines.

Intercepts: The (x)intercept is where the graph crosses the (x)axis, which is the point at which (y = 0). The (y)intercept is where the graph crosses the (y)axis, which is the point at which (x = 0).

Slope: The slope of a line is the ratio of the vertical change to the horizontal change between any two points on the line. The slopeintercept form of a line is (y = mx + b) where (m) is the slope and (b) is the (y)intercept.
Finding Equations of Lines
To write the equation of a line, we can use either the slopeintercept form or the pointslope form.

Slopeintercept form: This form is derived from the sloperise and run of the line. The slopeintercept form is (y = mx + b), where (m) is the slope and (b) is the (y)intercept.

Pointslope form: This form is derived from the slope of the line and a specific point on the line ((x_1, y_1)). The pointslope form is (y  y_1 = m(x  x_1)).
By understanding these concepts, you'll be able to analyze, sketch, and write the equations of transformed functions and their corresponding straight line graphs.
Learn about transformations of functions, including translations, reflections, and scalings, and their corresponding effects on straight line graphs. Explore how to interpret graphs by analyzing intercepts and slopes, and discover methods for finding equations of lines using slopeintercept and pointslope forms.
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