The x-intercept can be found by setting y = 0 and solving for x: 16 3, 0

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1 y=-3/4x+4 and y=2 x I need to graph the functions so I can clearly describe the graphs Specifically mention any key points on the graphs, including intercepts, vertex, or start/end points. What is the general shape of the graph? what is the domain and range for each equation? Is either equation a function or not? One of the graphs has been shifted three units upward and four units to the left. Discuss how this transformation affects the equation by rewriting the equation to incorporate those numbers. How do the words function, relation, domain, range, and transformation fit into working these problems? 1. y = 3 4 x + 4 This is the equation of a line having a slope of -3/4 and a y-intercept of (0, 4). This can be read directly from the equation since it is in the form y = mx + b, where m is the slope of the line, and b represents the y-coordinate of the y- intercept. The x-intercept can be found by setting y = 0 and solving for x: 0 = 3 4 x x = 4 x = x = 16 3 The coordinates of the x-intercept are then 16 3, 0. Since the graph is a line, there is no vertex, nor are there any start and end points. The domain of the graph is all real numbers. In interval notation, the domain is (-, ). The range of the graph is also all real numbers. In interval notation the range is also (-, ).

2 This equation is a function because there is one value of y for each value of x in the domain. In other words, the graph passes the vertical line test. The graph looks like this: 2. y = 2 x The graph of this equation will be a v-shaped graph consisting of two rays which meet at the origin. The x-intercept and the y-intercept both occur at the origin, (0, 0). The vertex of the graph occurs at the origin. There are no start or end points. The domain of the graph is all real numbers. In interval notation, the domain is (-, ). The range of the graph is all real numbers greater than or equal to 0. In interval notation, the range is [0, ). This equation is a function because there is one value of y for each value of x in the domain. In other words, the graph passes the vertical line test.

3 The graph looks like this: One of the graphs has been shifted three units upward and four units to the left. Discuss how this transformation affects the equation by rewriting the equation to incorporate those numbers. Applying this to the graph of y = 3 x + 4, and assuming that this equation 4 represents the equation after the transformation, we can find the original equation by reversing the transformation. A shift of three unit upward is achieved by adding a constant value of 3 to the original equation. To reverse this, subtract a constant of 3 to get: y = 3 4 x y = 3 4 x +1 A shift of 4 units to the right is achieved by substituting x 4 in place of x in the original equation. To reverse this, replace x in this equation with x + 4:

4 y = 3 ( 4 x + 4) +1 y = 3 4 x ( ) +1 y = 3 4 x 3+1 y = 3 4 x 2 The original equation, prior to the transformation would have been y = 3 4 x 2. On the other hand, if you want to start with the equation y = 3 x + 4 and make a 4 transformation by shifting the graph 3 units up and 4 units to the right, then you would follow these steps: Replace x with x 4. Add a constant of 3 to the equation This would give you: y = 3 ( 4 x 4) y = 3 4 x ( ) + 7 y = 3 4 x y = 3 4 x +10

5 How do the words function, relation, domain, range, and transformation fit into working these problems? A relation is an equation that defines how to calculate a value for the dependent variable, y, given a value for the independent variable, x. A function is a relation that has one and only one value of y for each value of x in the domain of the function. The domain of a function is the set of values for the independent variable for which the function is defined. Two common reasons for a function to not be defined at a particular value or interval would be that either the denominator of the function would take on a value of 0, or the value of a radicand (a value under the square root sign) would be negative. The domain is the set of the x- coordinates of every point contained on the graph of the function. The range of a function is the set of values produced by evaluating the function at each value in the domain. It would represent the set of the y-coordinates of every point contained on the graph of the function. A transformation is a modification of one function, which produces another function. A transformation can shift a graph in the y-direction (up or down), or in the x-direction (left or right). A transformation can also flip a graph about a horizontal axis by changing the sign of the x term. Finally, a transformation can also compress or stretch a graph by changing the value of the coefficient of the x term.