Algebra 1 Quiz 5.7 Through 5.10

The equation y = -4x + 2 describes a line with a slope of -4 and a y-intercept of 2. The slope of -4 means that for every increase of 1 in the x-coordinate, the y-coordinate decreases by 4. The y-intercept of 2 means that the line intersects the y-axis at the point (0, 2). Therefore, this equation accurately represents the line with the given slope and y-intercept.

The equation is in the form y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is 3 and the y-intercept is -6. This means that for every increase of 1 in the x-coordinate, the y-coordinate will increase by 3, and the line intersects the y-axis at -6.

The slope of a linear equation represents the rate of change of the dependent variable with respect to the independent variable. In this case, the slope is -1/4, which means that for every increase of 1 unit in the independent variable, the dependent variable decreases by 1/4 units.

The y-intercept is the value of the dependent variable when the independent variable is zero. In this case, the y-intercept is 3, which means that when the independent variable is zero, the dependent variable has a value of 3.

Therefore, the correct answer is slope: (-1/4); y-intercept: 3.

The slope of the graph is -2/3, which means that for every 3 units moved horizontally, the graph moves down 2 units vertically. The y-intercept is 2, which means that the graph intersects the y-axis at the point (0, 2).

The graph of f(x) = -3x - 2 is reflected across the y-axis, resulting in the graph of g(x) = 3x + 2. Additionally, the graph is translated 4 units up.

The equation y = 2x + 3 describes the line that contains the point (1,5) and has a slope of 2. This can be determined by substituting the coordinates of the point into the equation and verifying that it satisfies the equation. In this case, when x is 1, y is indeed 5, which confirms that the point lies on the line. Additionally, the coefficient of x in the equation is 2, which represents the slope of the line. Therefore, the equation y = 2x + 3 is the correct answer.

A line is perpendicular to another line if the slopes of the two lines are negative reciprocals of each other. The given equation y = 3x - 5 has a slope of 3. The equation y = (-1/3)x - 3 has a slope of -1/3, which is the negative reciprocal of 3. Therefore, y = (-1/3)x - 3 describes a line that is perpendicular to y = 3x - 5.

The equation y = 4x + 13 describes the line that passes through (-3,1) and is parallel to the line y = 4x + 1. Since the lines are parallel, they have the same slope, which is 4. The equation y = 4x + 13 has a slope of 4 and passes through the point (-3,1), making it the correct equation. The other options have different slopes or do not pass through the given point.

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The given equation y = -4x + 3 represents a line with a slope of -4. To find a line that is parallel to this, it must have the same slope of -4. Among the given options, the equation y = -4x + 2 has the same slope of -4 and therefore describes a line parallel to y = -4x + 3.

The equation y-5 = 4(x-3) goes through the point (3,5) because when we substitute x=3 and y=5 into the equation, both sides of the equation are equal. Additionally, the equation has a slope of 4, which means that for every increase of 1 in x, there is an increase of 4 in y.

The equation y = (1/2)x - 4 describes the line passing through (4, -2) with a slope of (1/2). The slope-intercept form of a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept. In this case, the slope is (1/2), so the equation starts with y = (1/2)x. The y-intercept is -4, which means the line passes through the point (0, -4). Since the line also passes through (4, -2), the equation is y = (1/2)x - 4.

The equation III can be rewritten as y = (3/2)x - 7. Since the coefficient of x is not the same as the other equations, it implies that the line represented by equation III is not parallel to the rest.

The equation y + 3 = 3(x - 4) describes the line through the points (4,-3) and (5,0). This equation is in point-slope form, where the slope is 3 and the point (4,-3) is on the line. By substituting the coordinates of the other point (5,0) into the equation, we can verify that it satisfies the equation. Therefore, this equation correctly represents the line passing through the given points.

The equation y = (-1/2)x - 5 describes a line that passes through the point (6, -8) and is perpendicular to the line described by 4x - 2y = 6. This is because the slope of the given line is -1/2, and the slope of a line perpendicular to it is the negative reciprocal of -1/2, which is 2. Therefore, the equation y = (-1/2)x - 5 represents a line with a slope of -1/2 and passes through the point (6, -8), satisfying the given conditions.

When the slope of a linear function changes to a smaller value, the graph becomes less steep. In this case, the original function f(x) = 3x + 7 has a slope of 3. If the slope changes to 2, it means that the graph will become less steep compared to the original graph. This means that the rate of change of the function will decrease, resulting in a gentler slope.

When a function is reflected across the y-axis, the sign of the coefficient of x changes. In the original function f(x) = -5x + 3, the coefficient of x is -5. So, when reflected across the y-axis, the new function h(x) will have the opposite sign, which is positive. Therefore, the correct answer is h(x) = 5x + 3.

The equation y = 5x + 14 describes the line with a slope of 5 and containing the point (-2, 4). This is because the equation is in the form y = mx + b, where m represents the slope and b represents the y-intercept. In this case, the slope is 5, indicating that for every increase of 1 in x, y will increase by 5. The y-intercept is 14, meaning that the line intersects the y-axis at the point (0, 14). Since the point (-2, 4) lies on this line, it satisfies the equation y = 5x + 14.