a The first thing we notice about this function is that it has a vertical asymptote at x=- 2 and a horizontal asymptote at y=0.
The vertical asymptote at x=-2 tells us that the function contains a fraction with an x in the denominator.
y=a/x-h+k
When substituting x=-2 into the function, it will lead to division by 0 which gives its vertical asymptote. This means h= -2.
y=a/x-( -2)+k ⇔ y=a/x+2+k
To determine the value of k, we have to think about the function's value when x goes towards large positive and large negative values. When x does this, then ax+2 will get continuously smaller. Therefore, the function's value approaches k.
|c|c|c|
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x & a/x+2+k & y [0.8em]
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[0.5em] -10 000 & a/- 10 000+2_(very small)+k & [0.5em] ≈ k [2.5em]
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[0.5em] 10 000 & a/10 000+2_(very small)+k & [0.5em] ≈ k [2.5em]
Because the graph has a horizontal asymptote at y=0, it must be that k= 0.
y=a/x+2+ 0 ⇔ y=a/x+2
To find the value of a, we must identify a point through which the graph passes. Let's find such a point in our diagram.
By substituting (-1,1) into the function, we can solve for a.
b Examining the diagram, we see that this is a parabola that opens upward and with a vertex on the y-axis at (0,-5). This means we can write the function in the following form.
y=ax^2-5
To determine if there is any vertical stretch, we have to find the value of a. For this purpose, we will identify a second point that is not the vertex.
We see that the function passes through (2,-1). By substituting this into the equation, we can solve for a.
To translate a graph in the positive horizontal direction by a units, we have to subtract a from the function's input.
y=(x-a)^3
Since the graph's inflection point is 3 units to the right of its parent graph's inflection point, we can write the function by subtracting 3 from x.
y=(x- 3)^3
d The graph shows an exponential function which means we can write it in the following format.
y=ab^x
Notice though that it has an horizontal asymptote that runs along y=-3.
To get an exponential function with a horizontal asymptote along y=-3, we have to translate the function in the negative vertical direction. In this case, we must subtract 3 from the function's output. If it didn't have this vertical shift, the function would intersect the y-axis at y=1 which means the initial value is a= 1.
y= 1b^x-3 ⇔ y=b^x-3
To find the value of b, we have to substitute a point that lies on the graph and solve for b. Let's identify such a point.
We see that the function passes through (2,1). By substituting this into the equation, we can solve for b.
Having found b=2, we can complete the function.
y=2^x-3
e The function describes a straight line which means we can write it in slope-intercept form.
y=mx+b
In this equation, m is the line's slope and b is the y-intercept. From the diagram, we can identify both of these values.
With these values, the equation of the line can be written as follows.
y=3x-6
f Like in Part C, this is a cubic function but it has been translated in both the horizontal and vertical direction. Let's first measure the horizontal distance and vertical distance between the inflection points.
This means we have to add 2 to the input and 3 to the output to make the necessary translations.
y=(x+2)^3+3
g This is a parabola which means we can describe it with a quadratic function. Since we can easily identify the parabola's vertex, we can write it in graphing form.
Graphing Form:& y=a(x-h)^2+k
Vertex:& (h,k)
Let's identify the parabola's vertex and another point on the graph which we will need to find the value of a.
By substituting h=-3 and k=-6 into the graphing form, we can start writing the equation.
y=a(x-(-3))^2+(-6)
⇕
y=a(x+3)^2-6
Next, to find the value of a, we must substitute the second point into the function and solve for a.
h Like in Part G, this is also a parabola where the vertex can be identified, immediately. Let's identify this point and a second point that also falls on the curve.
By substituting h=3 and k=6 into the graphing form of a parabola, we can start writing the equation.
y=a(x-3)^2+6
Next, to find the value of a, we must substitute the second point into the function and solve for a.
From here, we can complete the function as follows.
y=-(x-3)^2+6
i Like in Part C and F, this is a cubic function that has been translated in both the horizontal and vertical direction. Let's measure the horizontal and vertical distance between the inflection points of the given graph and its parent graph.
To translate it 3 units to the left, we add 3 to the input. To translate it 2 units down, we subtract from the output. With this information, we can write our function.
y=(x+3)^3-2
