Answer:
y=1, f(x) = -0.5, f(x) = 2
Step by Step Explanation:
For the first question:
(what is the value of the function when x = - 2)
When a question asks what the 'value of a function is', it is asking you to find the value of y.
So, this question is basically saying "if x = -2, what does y equal?". We can figure this out by finding what y is for the point (-2,y).
(remember that a point is written like (x,y), with x first and y second)
So, what point on the graph has x = -2? The answer is (-2,1). We know this because this graph goes through the point (-2,1). The value of the function is y = 1.
For the Second Question:
(the one where the first option is f(x) = 0.5x)
We can solve this question by finding the slope of the function. However, there is also a quick trick we can use to help narrow down our answers:
When we look at this function, the first thing we should notice is that it is decreasing. What is mean by this is that if you put your finger at the top of the function, and move it along the function, your finger is moving downwards.
So what does that tell us? It tells us that the slope is negative.
If a function is increasing, the slope will be a positive number
If a function is decreasing, the slope will be a negative number
This means that we automatically know the options F(x) = 2x and F(x) = 0.5x are wrong, since they both have a positive slope.
However, we still need to actually find the slope in order to answer the question. I am going to assume you understand what a slope is, but if not I recommend you watch a video explaining it as it is a very important concept.
Slope = Rise divided by Run
Rise: how much the y value changes
Run: how much the x value changes
I am going to use the points (0,0) and (2,-1), but you can use any 2 points you want in order to calculate the slope.
First let's find the rise:
in the first point, y = 0. In the second point, y = -1. So how much did y change? It changed by -1. Therefore the rise is -1 .
Now, let's find the run:
In the first point, x =0. In the second point, x=2. So how much did x change? It increased by 2. Therefore the run is 2.
We know the slope is rise divided by run, as I mentioned earlier. So the slope for this graph is -1/2 or -0.5 (if you convert to a decimal).
Therefore, the function that is graphed is f(x) = -0.5x
For the third question:
Again, we can solve this question by finding the slope.
Let's pick two points to use (you can choose any points), I'm going to pick (0,2) and (1,2).
I'll solve for the rise and run the same way I did before, by figuring out how much x and y changed between these two points.
The rise is 0, because y did not change at all (it was 2 in the first point, and 2 in the second point).
The run is 1, because x increased by 1.
Now, we can plug our answers in the slope formula:
Slope = rise/run = 0/1 = 0
The slope of this function is 0. What does that mean? It means that y never changes (because if you pick any 2 points on the graph, the difference between the y values will always be 0 (there will be no difference)).
In terms of the equation, it means there will be no x in our equation. Why? Because in a normal equation, you plug different values into x, and then you get a different value for y each time.
For example, the equation y = 2x + 1
if x = 3, then y = 2(3) + 1 = 7
if x = 4, then y = 2(4) + 1 = 9
notice how when x=3, y =7 but when x=4, y=9.
The value of x changes the value of y
But in this equation, y never changes. Therefore, there is no need to put an x into our equation.
So we know the answer cannot be f(x) = x-2 or f(x)=x+2 since they both have an x, but how do we know which other option to choose?
Well, let's go back to the equation for a line, y=mx+b. In this equation, m is the slope, and b is the y-intercept. We know that for this equation, there is no x, and our slope is 0, so our equation will be in the form of y=b.
What is the y-intercept? It is the point where our function touches the y-axis (the vertical line). This is 2, as we can see our function goes through the y-axis at 2.
Therefore, our equation will be f(x) = 2.
One thing you can remember, is that whenever your graph is a horizontal line, the slope will always be zero and the equation will never have an x in it, meaning all you have to do to find the equation is simply find the y-intercept.
If you have any questions, let me know!