The diagram below represents the graph of f(x)=a over x-p +q.T(5;3) determine the values of a,p, and q
Y=a over x-p +q y=a overx-4 +2 3=a over 5-4 +2 1=aover 1 a=1
To determine the values of a, p, and q for the graph of f(x)=a over x-p + q, given that T(5,3):
1. The point T(5,3) means that when x=5, f(x)=3. We can use this information to find the values of a, p, and q.
2. Plug in the values x=5 and f(x)=3 into the equation f(x)=a/(x-p) + q. This gives us:
3 = a/(5-p) + q.
3. Since we have two unknowns (a and p), we need another equation to solve for them. We can use the fact that the graph is a straight line (represented by a linear function).
4. In a linear function, the slope of the line represents the rate of change. Since the equation f(x)=a/(x-p) + q represents a linear function, the slope is given by -a/(x-p)^2.
5. To find the slope, we can differentiate the equation f(x) = a/(x-p) + q with respect to x:
f'(x) = -a/(x-p)^2.
6. Plug in the point T(5,3) into the derivative equation:
-a/(5-p)^2 = 3.
7. Solve the equation for a:
-a = 3(5-p)^2.
8. Simplify the equation:
-a = 3(25 - 10p + p^2).
9. Expand the equation:
-a = 75 - 30p + 3p^2.
10. Rearrange the equation:
a = -3p^2 + 30p - 75.
11. Now, substitute the expression for a obtained in step 10 into the equation from step 2:
3 = (-3p^2 + 30p - 75) / (5 - p) + q.
12. Simplify the equation:
3 = (-3p^2 + 30p - 75) / (5 - p) + q.
13. Multiply through by (5-p) to remove the denominator:
15 - 3p = -3p^2 + 30p - 75 + (5-p)q.
14. Rearrange the equation:
-3p^2 + 33p + (5-p)q - 90 = 0.
15. This equation can be solved for p and q. Once p and q are known, we can substitute them into the equation for a found in step 10 to obtain the value of a.
To determine the values of a, p, and q in the equation f(x) = a/(x - p) + q, given the point T(5, 3) on the graph, we can use the information provided.
The point T(5, 3) means that when x = 5, f(x) = 3. We can substitute these values into the equation and solve for a, p, and q.
Starting with the given equation: f(x) = a/(x - p) + q
Substituting x = 5 and f(x) = 3: 3 = a/(5 - p) + q
Now, we can rearrange the equation to solve for a, p, and q.
Step 1: Get rid of the fraction by multiplying both sides by (5 - p):
(5 - p) * 3 = a + q * (5 - p)
Step 2: Distribute to eliminate the parentheses:
15 - 3p = a + 5q - pq
Step 3: Rearrange the equation to isolate a:
a = 15 - 3p - 5q + pq
From here, we have expressed the value of a in terms of p and q. However, without additional information or constraints, we cannot determine the specific values of a, p, and q.