91Ó°ÊÓ

Linear functions are one of the simplest types of functions to understand and graph. The general form of a linear function is given by: \( f(x) = mx + b \). Here, 'm' represents the slope of the line and 'b' represents the y-intercept, where the line crosses the y-axis.

To graph a linear function, you only need two points. These points are often the x-intercept and y-intercept.
- The x-intercept is found by setting \( f(x) = 0 \) and solving for 'x'.
- The y-intercept is found by setting \( x = 0 \) and solving for 'f(x)'.
Once you have these two points, you plot them on the graph and draw a straight line through them.

For example, consider the function \( f(x) = 3x - 5 \).
- The y-intercept is found by setting \( x = 0 \): \( f(0) = -5 \). So the y-intercept is (0, -5).
- The x-intercept is found by setting \( f(x) = 0 \): \( 0 = 3x - 5 \) which gives \( x = \frac{5}{3} \). Thus, the x-intercept is \( \left( \frac{5}{3}, 0 \right) \).

Once plotted, a straight line through these points will give the graph of the linear function.

Quadratic functions graph to parabolas and have the general form of: \( f(x) = ax^2 + bx + c \), where 'a', 'b', and 'c' are constants.

The value of 'a' determines the direction of the parabola:
- If 'a' is positive, the parabola opens upwards.
- If 'a' is negative, the parabola opens downwards.

The graph of a quadratic function features a vertex, which is its highest or lowest point, and it might intersect the x-axis at two points (real & distinct roots), one point (a repeated root), or not at all (complex roots).

To accurately graph a quadratic function, you need:
- The y-intercept which is found by setting \( x = 0 \)
- The x-intercepts which are the roots of the equation \( f(x) = 0 \)
- The vertex, found using \( x = \frac{-b}{2a} \) and substituting back into the equation to get the y-value.

For instance, consider \( f(x) = -x^2 + 3x + 4 \). As 'a' is negative, this parabola opens downward. With the y-intercept (0, 4) and solving the quadratic equation, the x-intercepts are (-1, 0) and (4, 0).

Function intercepts are where the graph crosses the axes:

- The x-intercept(s) of a function is/are the point(s) where the function value (output) is zero. To determine x-intercepts, set \( f(x) = 0 \) and solve for 'x'. This will result in the points \( (x, 0) \).
- The y-intercept of a function is the point where the input is zero. To determine the y-intercept, set \( x = 0 \) and solve for 'f(x)' which results in the point \( (0, f(0)) \).

For example, in the linear function \( f(x) = 3x - 5 \), setting \( x = 0 \) gives us the y-intercept (0, -5). Setting \( f(x) = 0 \) and solving for x gives \( x = \frac{5}{3} \), so the x-intercept is \( \left( \frac{5}{3}, 0 \right) \).

In the quadratic function \( f(x) = -x^2 + 3x + 4 \), the y-intercept is (0, 4), found by setting \( x = 0 \). To find the x-intercepts, solve \( -x^2 + 3x + 4 = 0 \), which gives us the intercepts (-1, 0) and (4, 0).

The vertex of a parabola is the point where it reaches its maximum or minimum value, depending on the direction it opens.

In a quadratic function \( f(x) = ax^2 + bx + c \), the vertex can be determined using the formula \( x = \frac{-b}{2a} \).
- Once the x-coordinate of the vertex is found, substitute it back into the original equation to find the y-coordinate.

For instance, in the quadratic function \( f(x) = -x^2 + 3x + 4 \), the x-coordinate of the vertex is found with \( a = -1 \) and \( b = 3 \), yielding \( x = \frac{-3}{-2} = \frac{3}{2} \). Substituting \( x = \frac{3}{2} \) back into the function, the y-coordinate is \( f \left( \frac{3}{2} \right) = \frac{17}{4} \). Therefore, the vertex of this parabola is \( \left( \frac{3}{2}, \frac{17}{4} \right) \).

The vertex is a crucial point in graphing parabolas, as it gives a clear direction of the parabola's curvature, either opening upwards or downwards.