The equation you provided, \( y = \frac{1}{2}x + 1 \), is in the slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
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Slope: The slope of the line is given by the coefficient of \( x \), which in this case is \( \frac{1}{2} \). The slope indicates how steep the line is. A slope of \( \frac{1}{2} \) means that for every 1 unit increase in \( x \), \( y \) increases by \( \frac{1}{2} \) units. This results in a line that rises gently as it moves to the right.
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Angle: The angle of the line with respect to the positive x-axis can be determined from the slope. The relationship between slope \( m \) and angle \( \theta \) is given by the tangent function: \[ m = \tan(\theta) \] To find the angle, you can use: \[ \theta = \tan^{-1}(m) \] For your equation: \[ \theta = \tan^{-1}\left(\frac{1}{2}\right) \] This can be calculated to find the angle the line makes with the x-axis.
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Position vs. Angle: The position of the line is defined by its y-intercept \( b \) (which is 1 in your equation). This means that the line intersects the y-axis at (0, 1). Changing the slope (the steepness of the line) affects the angle with which it ascends or descends in relation to the x-axis but does not affect the y-intercept. Conversely, changing the y-intercept simply shifts the line up or down without changing its angle.
In summary:
- The slope determines the steepness and angle of the line.
- The y-intercept determines the position of the line on the y-axis.
- The angle can be calculated from the slope and indicates how "tilted" the line is.