Problem 30
Question
Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-2,-4)\) and \((1,-1)\)
Step-by-Step Solution
Verified Answer
The equation of the line in point-slope form is \(y + 4 = 1(x + 2)\) and in slope-intercept form it is \(y = x + 2\).
1Step 1: Calculate the Slope
The slope of the line passing through any given two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitue the given points \((-2,-4)\) and \((1,-1)\) into the formula to obtain the slope.
2Step 2: Write Point-slope Form
The equation in point-slope form is given by \(y - y_1 = m(x - x_1)\). Substitute the calculated slope \(m\) and any one set of the given coordinates into the equation.
3Step 3: Convert to Slope-Intercept Form
By simplifying the equation obtained in step 2, we get the equation in slope-intercept form which is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.
Key Concepts
Slope-Intercept Form EquationCalculating SlopeLine Equation from Two Points
Slope-Intercept Form Equation
Understanding the slope-intercept form equation is crucial for anyone learning algebra and coordinate geometry. It allows you to easily graph a straight line and understand its properties. The slope-intercept form is written as:
\( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis. This form is particularly useful because it provides a direct way to see both the angle of the line (its steepness) and where it intersects the y-axis, which are often important characteristics when graphing a linear relationship or comparing multiple lines.
\( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis. This form is particularly useful because it provides a direct way to see both the angle of the line (its steepness) and where it intersects the y-axis, which are often important characteristics when graphing a linear relationship or comparing multiple lines.
The Process of Converting to Slope-Intercept Form
Beginning with the point-slope form, which builds on the foundation of a known point and a slope, you can reorganize the equation by isolating \( y \) to one side. This involves distributing the slope value across the \( x \) and \( x_1 \) terms, and then adding or subtracting terms as needed to solve for \( y \) in terms of \( x \). Once you have isolated \( y \), the coefficient before \( x \) is your slope (\( m \)), and the constant term that remains is your y-intercept (\( b \)).Calculating Slope
The slope of a line is a measure of how steep it is and is a crucial concept in understanding the behavior of linear relationships. The formula to calculate the slope when you have two points is:
\( m = \frac{y_2 - y_1}{x_2 - x_1} \). The point \( (x_1, y_1) \) and the point \( (x_2, y_2) \) represent any two points on the line. The slope, denoted by \( m \), is the ratio of the change in the y-coordinates, or 'rise', to the change in the x-coordinates, or 'run'. When using this formula, be consistent with your points to avoid a sign error which could lead to the incorrect calculation of the slope's value.
\( m = \frac{y_2 - y_1}{x_2 - x_1} \). The point \( (x_1, y_1) \) and the point \( (x_2, y_2) \) represent any two points on the line. The slope, denoted by \( m \), is the ratio of the change in the y-coordinates, or 'rise', to the change in the x-coordinates, or 'run'. When using this formula, be consistent with your points to avoid a sign error which could lead to the incorrect calculation of the slope's value.
Interpreting Slope Values
- If the slope is positive, it means the line slopes upwards from left to right.
- A negative slope indicates that the line slopes downwards from left to right.
- If the slope is zero, the line is horizontal, indicating no change in y as x changes.
- An undefined or infinite slope corresponds to a vertical line where x does not change as y changes.
Line Equation from Two Points
When you have two points, you can not only calculate the slope but also write a complete equation of the line that passes through them. Begin by applying the slope formula:
\( m = \frac{y_2 - y_1}{x_2 - x_1} \), using the coordinates of the two given points. Once the slope is determined, choose either one of the two points as your reference point (\( x_1, y_1 \)) to substitute into the point-slope form equation:
\( y - y_1 = m(x - x_1) \).
Solve for \( y \) to rewrite this equation in the slope-intercept form for a more straightforward representation that confirms with the generalized slope-intercept form \( y = mx + b \).
\( m = \frac{y_2 - y_1}{x_2 - x_1} \), using the coordinates of the two given points. Once the slope is determined, choose either one of the two points as your reference point (\( x_1, y_1 \)) to substitute into the point-slope form equation:
\( y - y_1 = m(x - x_1) \).
Solve for \( y \) to rewrite this equation in the slope-intercept form for a more straightforward representation that confirms with the generalized slope-intercept form \( y = mx + b \).
Tips for Formulating the Line Equation
- Ensure accuracy in the slope calculation as any error will affect the entire line equation.
- Be mindful of negative signs when substituting values into the point-slope form. Incorrect signs can dramatically change the direction of your line.
- Always simplify your final equation as much as possible for easier graphing and interpretation.
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