Slope of a line and angle between two lines

In the standard form \(y=mx+c\), the coefficient \(m\) of \(x\) represents the slope. Here, \(y=-x\) can be written as \(y=(-1)x+0\), so the slope is \(-1\), which is negative. Therefore, the positive-slope option is incorrect. Exam tip: identify the coefficient of \(x\) directly to find the slope.

In slope-intercept form, \(y=mx+c\), the coefficient \(m\) is the slope. Comparing with \(y=2x+1\), the slope is 2; the constant 1 is the y-intercept.

In the line \(y=4\), the value of \(y\) remains constant for every \(x\), so the graph is horizontal. Using \(m=\frac{\Delta y}{\Delta x}\), we have \(\Delta y=0\), hence \(m=0\). The value 1 corresponds to a line rising 1 unit vertically for every 1 unit horizontally, not to this constant function. Exam tip: the graph of \(y=c\) is always horizontal and has slope 0.

In the linear form y = mx + c, the coefficient m of x is the slope of the graph. In f(x) = -2x + 4, the coefficient of x is -2, so the slope is -2. Option B, 2, has the wrong sign; a negative slope means that the function decreases as x increases. Exam tip: In a linear function written as y = mx + c, the coefficient of x is the slope.

For two points \((x_1,y_1)\) and \((x_2,y_2)\), the slope is \(m=\frac{y_2-y_1}{x_2-x_1}\). Thus, \(m=\frac{15-3}{6-2}=\frac{12}{4}=3\). Therefore, option B is correct. Option 4 is only the difference between the x-coordinates, not the slope. Exam tip: keep the order of subtraction the same in both the numerator and denominator.