Linear Functions and Slope Calculations
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Questions and Answers

What is the slope of the linear function $y = -4 - \frac{3}{2}x$?

  • -4
  • 0
  • -\frac{3}{2} (correct)
  • \frac{2}{3}

Which point does not lie on the graph of the function $y = -4 - \frac{3}{2}x$?

  • (4, 10) (correct)
  • (2, 7)
  • (0, -4)
  • (-2, -1)

What is the y-intercept of the function $y = -4 - \frac{3}{2}x$?

  • -4 (correct)
  • -3
  • 4
  • 0

If $x$ is increased by 4, what will be the change in $y$ based on the function $y = -4 - \frac{3}{2}x$?

<p>Decrease by 12 (D)</p> Signup and view all the answers

What is the domain of the function $y = -4 - \frac{3}{2}x$ if restricted to the table of values provided?

<p>x \in { -4, -2, 0, 2 } (A)</p> Signup and view all the answers

What is the slope of the line that passes through the points (0, -1) and (3, 2)?

<p>0.5 (C)</p> Signup and view all the answers

If the rise of the triangle on the graph is 3 units, what is the run based on the provided information?

<p>6 units (B)</p> Signup and view all the answers

Which of the following statements about the slope of the line is true?

<p>The slope indicates an uphill direction. (A)</p> Signup and view all the answers

What does the term 'rise' refer to in the context of slope calculations?

<p>The vertical change in the graph. (C)</p> Signup and view all the answers

Given the slope of 1/2, how would you interpret this in terms of rise and run?

<p>For every 1 unit of rise, there is 2 units of run. (B)</p> Signup and view all the answers

What is the slope calculated from the points (-6, -14) and (18, -20)?

<p>-1/4 (B)</p> Signup and view all the answers

Using the points (-8, 5) and (10, 15), what is the correct formula application to find the slope?

<p>slope = (15 - 5) / (10 - (-8)) (B)</p> Signup and view all the answers

What is the simplified result of the slope from the calculation slope = (15 - 5) / (10 - (-8))?

<p>5/9 (D)</p> Signup and view all the answers

If the coordinates (x₁, y₁) are (-8, 5) and (x₂, y₂) are (10, 15), which of the following describes a common mistake in calculating the slope?

<p>Mistaking the formula as (x₂ - x₁) / (y₂ - y₁) (D)</p> Signup and view all the answers

Which slope is steeper based on the examples provided?

<p>5/9 from points (-8, 5) and (10, 15) (C)</p> Signup and view all the answers

What is the slope of the line represented by the equation $y = 2x - 3$?

<p>2 (B)</p> Signup and view all the answers

What is the y-intercept of the line defined by the equation $y = 2x - 3$?

<p>-3 (D)</p> Signup and view all the answers

If a point on the line is (3, 0), what can be inferred about its significance?

<p>It is the x-intercept of the line. (D)</p> Signup and view all the answers

Given the slope of the line is 2, what is the change in y when x increases by 1?

<p>2 (A)</p> Signup and view all the answers

Which of the following equations represents a line with the same slope and y-intercept as the original line?

<p>$y = 2x - 3$ (B)</p> Signup and view all the answers

What is the standard form of the equation that corresponds to the x-intercept (6,0) and y-intercept (0,-4)?

<p>2x + y = 8 (D)</p> Signup and view all the answers

Which equation represents the same line as -8x - 4y = -16?

<p>2x + y = 4 (D)</p> Signup and view all the answers

What are the coordinates of the x-intercept given the intercept form derived from the equation -8x - 4y = -16?

<p>(6,0) (C)</p> Signup and view all the answers

If we rearrange -8x - 4y = -16 into slope-intercept form, what would the slope be?

<p>-2 (D)</p> Signup and view all the answers

What is the value of C in the equation Ax + By = C when the line passes through the points (6,0) and (0,-4)?

<p>-16 (C)</p> Signup and view all the answers

What is the effect of a slope that is a whole number on the graph of a linear function?

<p>It makes the graph steeper. (A)</p> Signup and view all the answers

In the point-slope form of a linear function, what does the variable 'm' represent?

<p>The rate of change in y with respect to x. (C)</p> Signup and view all the answers

Which of the following equations represents a linear function with a negative slope?

<p>y = -2x + 5 (B)</p> Signup and view all the answers

How does a fraction slope affect the steepness of a linear function's graph?

<p>It makes the graph flatter. (B)</p> Signup and view all the answers

What does the equation y - y₁ = m(x - x₁) signify in terms of graphing a linear function?

<p>It expresses the linear relationship between x and y at a specific point. (B)</p> Signup and view all the answers

What is the slope of the line represented by the equation 6x + 4y = 8 when converted to slope-intercept form?

<p>-3/2 (A)</p> Signup and view all the answers

What is the y-intercept of the line given by -15x - 3y = -6 once it is in slope-intercept form?

<p>2 (D)</p> Signup and view all the answers

When converting the equation 5y + 25x = 25 into slope-intercept form, what is the resulting simplified form?

<p>y = -5x + 5 (A)</p> Signup and view all the answers

What happens to the signs of the coefficients when converting from standard form to slope-intercept form if they are negative?

<p>They become positive in slope-intercept form (A)</p> Signup and view all the answers

If the equation 4x + 2y = 8 is converted to slope-intercept form, what is the final slope of the line?

<p>-2 (A)</p> Signup and view all the answers

What is the first step in converting the point-slope form equation y - 8 = $ rac{1}{2}$(x + 2) to slope-intercept form?

<p>Distribute $ rac{1}{2}$ into the equation (C)</p> Signup and view all the answers

What is the value of the y-intercept in the equation y = $ rac{1}{2}$x + 9?

<p>9 (D)</p> Signup and view all the answers

What is the slope of the line represented by the equation y = -3x + 5?

<p>-3 (B)</p> Signup and view all the answers

After simplifying y - 12 = -3(x + 5/3) - 2, what is the resulting expression for y?

<p>y = -3x + 5 (C)</p> Signup and view all the answers

In the context of converting point-slope form to slope-intercept form, what does 'm' represent?

<p>The slope of the line (A)</p> Signup and view all the answers

What is the slope of the equation represented in slope-intercept form from Equation 4 (3y = -10)?

<p>-1.5 (D)</p> Signup and view all the answers

What would be the value of y in Equation 5 (-3x + y = ?) if x = 6?

<p>18 (A)</p> Signup and view all the answers

Which equation, when converted to slope-intercept form, has a y-intercept of 24?

<p>2x + 3y = 24 (D)</p> Signup and view all the answers

What is the first step to convert Equation 2 (48 - 4y = 76 - 48) to slope-intercept form?

<p>Simplify the right side (A)</p> Signup and view all the answers

After converting Equation 1 (3x - x + 48 = 3) to slope-intercept form, what will be the resulting equation?

<p>y = -2x + 45 (D)</p> Signup and view all the answers

What is the equation of the line parallel to $y=3x+8$ that passes through the point $(-3, 7)$?

<p>$y=3x+4$ (C)</p> Signup and view all the answers

Which of the following pairs of slopes indicate that two lines are perpendicular?

<p>$ rac{3}{4}$ and $- rac{4}{3}$ (A)</p> Signup and view all the answers

If a line has the equation $y = - rac{2}{5}x + 5$, which of the following equations represents a line that is parallel to it?

<p>$y = - rac{2}{5}x - 3$ (D)</p> Signup and view all the answers

Which equation correctly illustrates two lines that are perpendicular to each other?

<p>$y = rac{1}{3}x - 1$ and $y = -3x + 5$ (D)</p> Signup and view all the answers

What can be concluded about the slopes of the lines given by the equations $y = -7x + 2$ and $y = 0.14x - 5$?

<p>The lines are parallel. (A)</p> Signup and view all the answers

Flashcards

Function Rule

An equation that defines the relationship between input (x) and output (y) values.

Coordinate Pairs

A pair of numbers (x, y) that represent a point on a graph.

Table of Values

A table that lists different input (x) values and their corresponding output (y) values according to a function rule.

Graphing Linear Functions

Plotting points that represent a linear equation and connecting them with a straight line.

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Linear Function

A function whose graph is a straight line. It can be represented by the equation y = mx + b.

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Slope

The steepness of a line, calculated by the ratio of vertical change (rise) to horizontal change (run).

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Rise

The vertical change between two points on a line, representing the increase or decrease in the y-coordinate.

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Run

The horizontal change between two points on a line, representing the increase or decrease in the x-coordinate.

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Downhill Slope

A slope with a negative value, indicating a downward direction.

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Uphill Slope

A slope with a positive value, indicating an upward direction.

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Slope Formula

The formula used to calculate the slope of a line given two points (x₁, y₁) and (x₂, y₂): slope = (y₂ - y₁) / (x₂ - x₁)

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Slope Calculation

The process of substituting the coordinates of two given points into the slope formula and simplifying the result to obtain the slope of the line

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Example 1

The slope of a line passing through points (-8, 5) and (10, 15) is calculated as (15 - 5) / (10 - (-8)) = 5/9

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Example 2

The slope of a line passing through points (-6, -14) and (18, -20) is calculated as (-20 - (-14)) / (18 - (-6)) = -1/4

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What is slope?

Slope represents the steepness or incline of a line. It indicates how much the y-value changes for every unit change in the x-value

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Slope: Flatter

A line with a fraction as its slope will appear flatter than a line with a whole number as its slope.

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Slope: Steeper

A line with a whole number as its slope will appear steeper than a line with a fraction as its slope.

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Slope: Positive

A line with a positive slope will slant upwards from left to right.

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Point-Slope Form

An equation of a line that uses a specific point (x₁, y₁) on the line and the slope (m) to define the relationship between x and y.

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Slope-Intercept Form

An equation of a line that directly shows the slope (m) and the y-intercept (b).

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Convert Point-Slope to Slope-Intercept

Transforming an equation from point-slope form to slope-intercept form by isolating the 'y' variable.

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What is the y-intercept?

The point where the line crosses the y-axis. It is represented by 'b' in the equation y = mx + b.

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What is the slope?

The steepness of a line. It is represented by 'm' in the equation y = mx + b.

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Y-Intercept

The point where a line crosses the y-axis. It's represented by the value 'b' in the slope-intercept form (y = mx + b).

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X-Intercept

The point where a line crosses the x-axis. It's the point where the y-value is 0.

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How to find the Slope

The slope of a line can be calculated using any two points on the line. Divide the difference in y-values (vertical rise) by the difference in x-values (horizontal run).

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Standard Form

A way to express a linear equation as Ax + By = C, where A, B, and C are constants.

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Find A, B, and C

Using the x-intercept and y-intercept, substitute the x and y values into the standard form equation (Ax + By = C) to create two equations. Solve the system of equations to find A, B, and C.

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Write the Linear Equation

Once A, B, and C are found, plug them back into the standard form equation (Ax + By = C) to write the complete linear equation.

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Convert to Slope-Intercept

Rearranging a linear equation from standard form to slope-intercept form.

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x-term Sign Change

When converting from standard form to slope-intercept form, the x-term's sign flips.

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Coefficient Division

The coefficient of 'y' in standard form divides both sides of the equation during conversion to slope-intercept form.

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Isolate y

The process of rearranging an equation to have y by itself on one side.

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Convert to Slope-Intercept Form

Transforming an equation from standard form (Ax + By = C) to slope-intercept form (y = mx + b).

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Parallel Lines

Lines that have the same slope but different y-intercepts. They never intersect.

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Perpendicular Lines

Lines that intersect at a right (90-degree) angle. Their slopes are negative reciprocals of each other.

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Finding the Equation of a Parallel Line

To find the equation of a line parallel to a given line, use the same slope but substitute the given point into the point-slope form.

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