Questions: 45-50 Find an expression for the function whose graph is the given curve. 45. The line segment joining the points (1,-3) and (5,7) 46. The line segment joining the points (-5,10) and (7,-10) 47. The bottom half of the parabola x+(y-1)^2=0

Solution Steps

To find the expressions for the given functions, we need to determine the equations of the lines or curves described.

The equation of a line can be found using the two-point form of the line equation.

Similarly, we use the two-point form of the line equation to find the equation of this line segment.

To find the expression for the bottom half of the parabola, we need to solve the given equation for \( y \) and consider only the negative square root to get the bottom half.

To find the equation of the line passing through the points \((1, -3)\) and \((5, 7)\), we use the two-point form of the line equation:

\[ y = mx + b \]

where \( m \) is the slope and \( b \) is the y-intercept. The slope \( m \) is calculated as:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - (-3)}{5 - 1} = \frac{10}{4} = 2.5 \]

The y-intercept \( b \) is found using one of the points, say \((1, -3)\):

\[ -3 = 2.5 \cdot 1 + b \implies b = -3 - 2.5 = -5.5 \]

Thus, the equation of the line is:

\[ y = 2.5x - 5.5 \]

Similarly, for the points \((-5, 10)\) and \((7, -10)\), the slope \( m \) is:

\[ m = \frac{-10 - 10}{7 - (-5)} = \frac{-20}{12} = -1.6667 \]

Using the point \((-5, 10)\) to find the y-intercept \( b \):

\[ 10 = -1.6667 \cdot (-5) + b \implies b = 10 - 8.3333 = 1.6667 \]

Thus, the equation of the line is:

\[ y = -1.6667x + 1.6667 \]

To find the bottom half of the parabola given by the equation \(x + (y-1)^2 = 0\), we solve for \( y \):

\[ x + (y-1)^2 = 0 \implies (y-1)^2 = -x \implies y - 1 = \pm \sqrt{-x} \]

For the bottom half, we take the negative square root:

\[ y - 1 = -\sqrt{-x} \implies y = 1 - \sqrt{-x} \]

Final Answer