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Zorluk: KolayLinear Functions and Graphs

In the xyxy-plane, the graph of the linear function hh has a slope of 33 and passes through the point (5,8)(5, 8). What is the value of h(11)h(11)?

Cevap: 26

Cevap

The value of h(11)h(11) is 26.
The correct answer is 26. Since the function hh is linear, its rate of change (slope) is constant. A slope of 3 means that for every 1-unit increase in xx, the value of h(x)h(x) increases by 3. The change in xx from 5 to 11 is 11−5=611 - 5 = 6. Therefore, the corresponding increase in h(x)h(x) is 3×6=183 \times 6 = 18. Adding this increase to the known value of h(5)=8h(5) = 8 gives h(11)=8+18=26h(11) = 8 + 18 = 26.

Adım Adım Çözüm

1
Determine the change in the input variable xx from the known point to the target point.
The change in xx is 11−5=611 - 5 = 6.
This identifies how many units the input increases from x=5x = 5 to x=11x = 11.
2
Multiply the change in xx by the slope to find the corresponding change in the function value.
The change in the function value is 3×6=183 \times 6 = 18.
The slope represents the constant rate of change (change in yy divided by change in xx). Therefore, multiplying the slope by the change in xx gives the total change in the output.
3
Add the calculated change in the function value to the initial function value.
h(11)=8+18=26h(11) = 8 + 18 = 26.
Adding the total increase in output to the starting output value at x=5x = 5 yields the value of the function at x=11x = 11.

Anahtar Kavram

Linear functions have a constant rate of change, which is represented by the slope. The change in the output value is equal to the slope multiplied by the change in the input value.

Alternatif Yöntem

Alternatively, find the equation of the line using the point-slope form: y−y1=m(x−x1)y - y_1 = m(x - x_1). Substituting the slope m=3m = 3 and the point (5,8)(5, 8) gives y−8=3(x−5)y - 8 = 3(x - 5), which simplifies to y=3x−7y = 3x - 7. Thus, the function is defined by h(x)=3x−7h(x) = 3x - 7. Evaluating this function at x=11x = 11 yields h(11)=3(11)−7=33−7=26h(11) = 3(11) - 7 = 33 - 7 = 26.
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Linear Functions and Graphs

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