Yes, $y = 2x + 5$ is a function. For every input value of $x$, there is exactly one unique output value of $y$. This relationship satisfies the definition of a function, as it passes the vertical line test and can be classified as a linear function.
To graph y = 2x + 5, start by plotting the y-intercept, which is (0, 5). From that point, use the slope, which is 2 (or 2/1). This means for every 1 unit you move to the right on the x-axis, you move 2 units up on the y-axis. Plot a few more points and then draw a straight line through them. Alternatively, you can pick two x values, calculate their corresponding y values, plot those two points, and connect them.
The slope of the linear equation y = 2x + 5 is 2. In the standard slope-intercept form, y = mx + b, 'm' represents the slope of the line. In this specific equation, the coefficient of x is 2, indicating that for every unit increase in x, y increases by 2 units. This positive slope means the line rises from left to right.
The y-intercept of the equation y = 2x + 5 is 5. In the slope-intercept form, y = mx + b, 'b' represents the y-intercept, which is the point where the line crosses the y-axis. When x is 0, y equals 5, so the coordinates of the y-intercept are (0, 5). This is a crucial point for graphing the line.
To find the x-intercept of y = 2x + 5, you set y equal to 0 and then solve for x. So, 0 = 2x + 5. Subtract 5 from both sides to get -5 = 2x. Then, divide by 2 to find x = -5/2 or -2.5. The x-intercept is therefore (-2.5, 0), which is the point where the line crosses the x-axis.
Yes, y = 2x + 5 is definitively a linear equation. It fits the standard form y = mx + b, where m and b are constants. When graphed, it produces a straight line. The variables x and y are raised only to the power of 1, and there are no products of variables, square roots of variables, or variables in denominators, all characteristics of linear equations.
The domain of y = 2x + 5 is all real numbers. Since this is a linear function, there are no restrictions on the values that x can take. You can substitute any real number for x, and you will always get a valid real number for y. This is typical for all polynomial functions, including linear ones.
The range of y = 2x + 5 is also all real numbers. Because the function is linear and its slope is not zero, y can take on any real value. As x spans all real numbers, y will similarly span all real numbers from negative infinity to positive infinity. There are no upper or lower bounds for the output values.
To solve for x in the equation y = 2x + 5, you need to isolate x. First, subtract 5 from both sides of the equation: y - 5 = 2x. Then, divide both sides by 2: (y - 5) / 2 = x. So, x = (y - 5) / 2. This expression allows you to find x for any given value of y.
Changing the '5' in y = 2x + 5 directly affects the y-intercept of the graph. The '5' represents the 'b' value in the slope-intercept form (y = mx + b). If you increase the '5', the entire line shifts vertically upwards. If you decrease it, the line shifts vertically downwards, while maintaining the same slope of 2.