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Chapter 13: Problem 47

Which of the intercepts of \(f(x)=-\frac{9}{10} x+9\) is closer to the origin?

Short Answer

Expert verified
The y-intercept is closer to the origin.

Step by step solution

01

- Identify x-intercept

To find the x-intercept, set the function equal to zero and solve for x: \[ f(x) = -\frac{9}{10} x + 9 \] Set \( f(x) = 0 \): \[ 0 = -\frac{9}{10}x + 9 \] Solve for x: \[ -9 = -\frac{9}{10}x \] \[ x = 10 \]
02

- Identify y-intercept

To find the y-intercept, evaluate the function when x is zero: \[ f(x) = -\frac{9}{10}x + 9 \] Set \( x = 0 \): \[ f(0) = -\frac{9}{10}(0) + 9 \] \[ f(0) = 9 \]
03

- Calculate distance to the origin

Calculate the distance from the origin to each intercept. For the x-intercept (10,0): \[ \sqrt{10^2 + 0^2} = 10 \] For the y-intercept (0,9): \[ \sqrt{0^2 + 9^2} = 9 \]
04

- Compare distances

Compare the distances calculated: 10 units for the x-intercept and 9 units for the y-intercept. The y-intercept is closer to the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-intercept
The x-intercept of a function is the point where the graph crosses the x-axis. To find the x-intercept, we need to set the function equal to zero and solve for x. For example, consider the function: \( f(x) = -\frac{9}{10}x + 9 \). When we set \( f(x) = 0 \), the equation becomes \( 0 = -\frac{9}{10}x + 9 \). Solving for x, we get: \( -9 = -\frac{9}{10}x \). \
\
By isolating x, we find \( x = 10 \). So, the x-intercept of the function \( f(x) = -\frac{9}{10}x + 9 \) is at the coordinate (10, 0). \
    \
  • Step 1: Set \( f(x) = 0 \).
    • \
  • Step 2: Solve the equation for x.
    • \
  • Step 3: The x-intercept is the point (x, 0).
    • \
\
In essence, the x-intercept indicates where the function's value is zero along the x-axis.
y-intercept
The y-intercept is where the graph of the function crosses the y-axis. To determine this, evaluate the function at \( x = 0 \). For the function \( f(x) = -\frac{9}{10}x + 9 \), set \( x = 0 \) to find the y-intercept: \( f(0) = -\frac{9}{10}(0) + 9 \). This simplifies to \( f(0) = 9 \).\
\
Therefore, the y-intercept of the function \( f(x) = -\frac{9}{10}x + 9 \) is at the point (0, 9).\
\
To summarize the steps: \
\
    \
  • Set x to 0 in the function.
    • \
  • Compute the resulting value of the function at this x value.
    • \
  • The y-intercept is the point (0, f(0)).
    • \
\
The y-intercept is a crucial characteristic because it represents the function's value when x is zero.
distance to origin
To compare the distances of intercepts from the origin, we use the distance formula. The origin is the point (0, 0) on a graph. The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \].\
\
For the x-intercept (10, 0), the distance from the origin is \( \sqrt{(10 - 0)^2 + (0 - 0)^2} = \sqrt{10^2} = 10 \) units.\
\
For the y-intercept (0, 9), the distance from the origin is \( \sqrt{(0 - 0)^2 + (9 - 0)^2} = \sqrt{9^2} = 9 \) units.\
\
    \
  • Step 1: Use the distance formula for each intercept.
    • \
  • Step 2: Calculate the distances.
    • \
  • Step 3: Compare the results.
    • \
\
In our example, the y-intercept at (0, 9) is closer to the origin than the x-intercept at (10, 0), since 9 units is less than 10 units.

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