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Zorluk: Çok zorLinear Functions and Graphs

A linear function ff is defined by f(x)=mx+bf(x) = mx + b, where mm and bb are constants. The graph of ff in the xyxy-plane passes through the point (2,5)(2, 5). A second linear function gg is defined by g(x)=f(x−3)+4g(x) = f(x - 3) + 4. The graph of gg has an xx-intercept of (k,0)(k, 0) and a yy-intercept of (0,−k)(0, -k), where kk is a nonzero constant. What is the value of f(8)f(8)?

Cevap: 11

Cevap

11
The correct answer is 11. By using the given point on the graph of f(x)f(x), we express its y-intercept bb in terms of its slope mm as b=5−2mb = 5 - 2m. Substituting this expression into the translation equation g(x)=f(x−3)+4g(x) = f(x - 3) + 4 yields g(x)=mx−5m+9g(x) = mx - 5m + 9. Setting the y-intercept of g(x)g(x), which is −5m+9-5m + 9, equal to −k-k gives k=5m−9k = 5m - 9. Evaluating g(x)g(x) at its x-intercept x=kx = k gives mk−5m+9=0mk - 5m + 9 = 0. Substituting the relation −5m+9=−k-5m + 9 = -k into this equation yields mk−k=0mk - k = 0, or k(m−1)=0k(m - 1) = 0. Since kk is nonzero, we find m=1m = 1, which in turn gives b=3b = 3. Thus, the linear function is f(x)=x+3f(x) = x + 3, and f(8)=11f(8) = 11.

Adım Adım Çözüm

1
Express the y-intercept constant bb in terms of the slope mm.
b=5−2mb = 5 - 2m
Since the graph of f(x)=mx+bf(x) = mx + b passes through (2,5)(2, 5), substituting x=2x = 2 and f(x)=5f(x) = 5 gives 5=2m+b5 = 2m + b.
2
Write the expression for g(x)g(x) in terms of mm.
g(x)=mx−5m+9g(x) = mx - 5m + 9
By definition, g(x)=f(x−3)+4=m(x−3)+b+4g(x) = f(x - 3) + 4 = m(x - 3) + b + 4. Substituting b=5−2mb = 5 - 2m simplifies the expression to g(x)=mx−5m+9g(x) = mx - 5m + 9.
3
Relate the y-intercept of g(x)g(x) to the parameter kk.
k=5m−9k = 5m - 9
The y-intercept of g(x)g(x) is g(0)=−5m+9g(0) = -5m + 9. Since the y-intercept is given as (0,−k)(0, -k), we set −k=−5m+9-k = -5m + 9, which gives k=5m−9k = 5m - 9.
4
Set up an equation using the x-intercept of g(x)g(x).
g(k)=mk−5m+9=0g(k) = mk - 5m + 9 = 0
Since the x-intercept of g(x)g(x) is (k,0)(k, 0), substituting x=kx = k into the expression for g(x)g(x) must yield 00.
5
Solve for the slope mm.
m=1m = 1
Substituting −5m+9=−k-5m + 9 = -k into the equation mk−5m+9=0mk - 5m + 9 = 0 gives mk−k=0mk - k = 0, which factors as k(m−1)=0k(m - 1) = 0. Since kk is a nonzero constant, we divide by kk to get m−1=0m - 1 = 0, so m=1m = 1.
6
Find the constant bb and write the final formula for f(x)f(x).
f(x)=x+3f(x) = x + 3
Substituting m=1m = 1 back into b=5−2mb = 5 - 2m yields b=3b = 3. Therefore, f(x)=x+3f(x) = x + 3.
7
Evaluate f(8)f(8).
11
Substituting x=8x = 8 into the function f(x)=x+3f(x) = x + 3 gives f(8)=8+3=11f(8) = 8 + 3 = 11.

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Linear function transformations and intercept properties
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Linear Functions and Graphs

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