Math  /  Algebra

QuestionThe graph below is on a semi-log scale, as indicated.
Find an equation for the graph shown y(x)=y(x)= \square

Studdy Solution

STEP 1

1. The graph is on a semi-logarithmic scale, meaning the y-axis is logarithmic.
2. The equation of the line is in the form ln(y)=mx+b \ln(y) = mx + b because the y-axis is ln(y) \ln(y) .
3. The line passes through the points (5,5)(-5, -5) and (5,5)(5, 5).

STEP 2

1. Determine the slope m m of the line.
2. Use one of the points to find the y-intercept b b .
3. Write the equation in terms of ln(y) \ln(y) .
4. Convert the equation to solve for y(x) y(x) .

STEP 3

Calculate the slope m m of the line using the formula for slope:
m=y2y1x2x1 m = \frac{y_2 - y_1}{x_2 - x_1}
Substitute the given points (5,5)(-5, -5) and (5,5)(5, 5):
m=5(5)5(5)=5+55+5=1010=1 m = \frac{5 - (-5)}{5 - (-5)} = \frac{5 + 5}{5 + 5} = \frac{10}{10} = 1

STEP 4

Use the slope m=1 m = 1 and one of the points, say (5,5)(-5, -5), to find the y-intercept b b using the equation:
ln(y)=mx+b \ln(y) = mx + b
Substitute the point and slope:
5=1(5)+b -5 = 1(-5) + b
5=5+b -5 = -5 + b
Solve for b b :
b=0 b = 0

STEP 5

Write the equation of the line in terms of ln(y) \ln(y) :
ln(y)=1x+0 \ln(y) = 1 \cdot x + 0
Simplify:
ln(y)=x \ln(y) = x

STEP 6

Convert the equation to solve for y(x) y(x) by exponentiating both sides to remove the natural logarithm:
y=ex y = e^{x}
Thus, the equation for the graph is:
y(x)=ex y(x) = e^{x}

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