Question

1. is the relation shown below a function? use the graph below to justify your answer. (0, 3), (1, 4), (2, 3), (3, 0), (5, 4) graph of a relation 2. consider the two functions below. which one of these functions is linear? what is its equation? enter any answers to two decimal places. function a graph of a line function b table with x: 1,2,3,4,5 and y:1,8,27,64,125 function is linear. its equation is y = x + . 3. which function has a greater rate of change? function a: y = 10x - 3 function b: table with x:1,2,3,4,5 and y:20,15,10,5,0 function a has a rate of change of and function b has a rate of change of , so function has a greater rate of change. 4. how would you describe the graph of the function in interval 2? graph of a piecewise function a) the graph of the function is increasing. b) the graph of the function is decreasing. c) the graph of the function is constant. d) the slope is positive.

Explanation:

Question 1

Step1: Recall the definition of a function

A function is a relation where each input (x - value) has exactly one output (y - value).

Step2: Check the given ordered pairs

The ordered pairs are \((0,3),(1,4),(2,3),(3,0),(5,4)\). For each \(x\) - value (0, 1, 2, 3, 5), there is only one \(y\) - value. Even though the \(y\) - value 3 is associated with \(x = 0\) and \(x=2\), and \(y = 4\) is associated with \(x = 1\) and \(x = 5\), the definition of a function requires that each \(x\) has only one \(y\), not that each \(y\) has only one \(x\). So we check the \(x\) - values: 0 has 3, 1 has 4, 2 has 3, 3 has 0, 5 has 4. No \(x\) - value is repeated with different \(y\) - values.

Step1: Recall the definition of a linear function

A linear function has a constant rate of change (slope) and its graph is a straight line. The general form is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.

Step2: Analyze Function A

Function A is a straight line. To find the slope \(m\), we can use two points. From the graph, when \(x = 0\), \(y=1\) (wait, looking at the graph, when \(x = 0\), \(y\) - intercept seems to be 1? Wait, the graph passes through \((0,1)\) and \((1,2)\)? Wait, no, looking at the grid: when \(x=- 2\), \(y = - 1\); \(x = 0\), \(y=1\); \(x = 2\), \(y = 3\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Using \((0,1)\) and \((2,3)\), \(m=\frac{3 - 1}{2-0}=\frac{2}{2}=1\). Wait, maybe the correct points: from the graph, the line passes through \((-2,-1)\) and \((0,1)\). Then \(m=\frac{1-(-1)}{0 - (-2)}=\frac{2}{2} = 1\). And the y - intercept \(b\) (when \(x = 0\)) is 1. Wait, but let's check the table for Function B. Function B has \(x = 1,y = 1\); \(x = 2,y = 8\); \(x=3,y = 27\); \(x = 4,y = 64\); \(x=5,y = 125\). Notice that \(y=x^{3}\) (since \(1^{3}=1\), \(2^{3}=8\), \(3^{3}=27\), etc.), which is a cubic function, not linear. So Function A is linear.

Step3: Find the equation of Function A

Using the slope - intercept form \(y=mx + b\). From the graph, when \(x = 0\), \(y = 1\) (y - intercept \(b = 1\)). Let's find the slope between two points. Let's take \((0,1)\) and \((1,2)\), slope \(m=\frac{2 - 1}{1-0}=1\). Wait, but looking at the grid again, if we take \((-2,-1)\) and \((0,1)\), \(m=\frac{1-(-1)}{0-(-2)} = 1\). So the equation is \(y=x + 1\)? Wait, maybe the graph in the problem has a different slope. Wait, the graph of Function A: when \(x=-4\), \(y=-3\); \(x=-2\), \(y=-1\); \(x = 0\), \(y = 1\); \(x = 2\), \(y = 3\); \(x = 4\), \(y = 5\). So the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-1-(-3)}{-2-(-4)}=\frac{2}{2}=1\), and \(b = 1\). So the equation is \(y = 1x+1\) or \(y=x + 1\). But maybe the intended slope is 1.5? Wait, no, let's check the points again. If the line passes through \((0,1)\) and \((2,4)\), then slope \(m=\frac{4 - 1}{2-0}=\frac{3}{2}=1.5\) and \(b = 1\). Wait, maybe my initial point - picking was wrong. Let's use the two - point formula. Let's take \((-2, - 2)\) and \((2,4)\). Then \(m=\frac{4-(-2)}{2-(-2)}=\frac{6}{4}=1.5\) and when \(x = 0\), from the graph, \(y = 1\)? No, when \(x = 0\), the y - intercept: looking at the graph, the line crosses the y - axis at \(y = 1\)? Wait, maybe the correct way is to use the formula for slope between two points on the line. Let's take \((0,1)\) and \((2,4)\): \(m=\frac{4 - 1}{2-0}=\frac{3}{2}=1.5\), \(b = 1\). So the equation is \(y = 1.5x+1\).

Step1: Recall the formula for rate of change (slope) of a linear function

For a linear function in the form \(y=mx + b\), the rate of change (slope) \(m\) is the coefficient of \(x\). For a function given by a table, the rate of change \(m=\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Find the rate of change of Function A

Function A is \(y = 10x-3\), so its rate of change (slope) is \(m = 10\).

Step3: Find the rate of change of Function B

For Function B, we use the table values. Let's take two points, say \((1,20)\) and \((2,15)\). Then \(\Delta y=15 - 20=- 5\), \(\Delta x=2 - 1 = 1\). So the rate of change \(m=\frac{-5}{1}=-5\). The magnitude of the rate of change of Function A (10) is greater than the magnitude of the rate of change of Function B (5, but actually - 5). So Function A has a greater rate of change. The initial answer had Function A's rate of change as 3 (incorrect) and Function B's rate of change as 5 (incorrect).

Answer:

Yes, the relation is a function because each input \(x\) - value has exactly one output \(y\) - value. The initial answer "No because It has two outputs" is incorrect.

Question 2