Exercise: Linear Functions and Their Properties

Problem Statement:

Given the function $f(x) = 3x + 9$:

1. (a) Describe the shape of the graph of the function and find algebraically the points where it intersects the X and Y axes.
2. (b) Find algebraically the point of intersection of the graphs of $f(x)$ and $g(x) = -x + 11$.
3. (c) If a line passes through the point $(2, 1)$ and has a slope of $-2$, determine its equation.
4. (d) Prove that the line passing through the points $(-2, -8)$ and $(2, 4)$ is parallel to the graph of the function $\phi(x) = 3x + 1$.

Blog Post: Understanding Linear Functions with Worked Examples

Linear functions are fundamental in mathematics, representing relationships with a constant rate of change. In this blog post, we will explore and solve a set of problems to understand their properties and applications.

(a) Shape and Intercepts of $f(x) = 3x + 9$

The function $f(x) = 3x + 9$ is a linear equation, representing a straight line. The slope ($m$) of the line is $3$, and the y-intercept ($b$) is $9$.

To find the intercepts:

• X-axis: Set $f(x) = 0$

  $$3x + 9 = 0 \implies x = -3$$Thus, the X-intercept is $(-3, 0)$.

• Y-axis: Set $x = 0$

  $$f(0) = 3(0) + 9 = 9$$Thus, the Y-intercept is $(0, 9)$.

Graphically, the line crosses the X-axis at $(-3, 0)$ and the Y-axis at $(0, 9)$.

(b) Intersection of $f(x) = 3x + 9$ and $g(x) = -x + 11$

To find the intersection point, set $f(x) = g(x)$:

$$3x + 9 = -x + 11$$ $$4x = 2 \implies x = \frac{1}{2}$$

Substitute $x = \frac{1}{2}$ into either equation, e.g., $f(x)$:

$$f\left(\frac{1}{2}\right) = 3\left(\frac{1}{2}\right) + 9 = \frac{3}{2} + 9 = \frac{21}{2}$$

Thus, the intersection point is $\left(\frac{1}{2}, \frac{21}{2}\right)$.

(c) Equation of a Line through $(2, 1)$ with Slope $-2$

The slope-intercept form of a line is $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept.

Using the point $(2, 1)$:

$$1 = -2(2) + c \implies 1 = -4 + c \implies c = 5$$

Thus, the equation is:

$$y = -2x + 5$$

(d) Proving Parallelism of Two Lines

Two lines are parallel if they have the same slope.

• The slope of the line passing through $(-2, -8)$ and $(2, 4)$:

  $$m = \frac{4 – (-8)}{2 – (-2)} = \frac{4 + 8}{2 + 2} = \frac{12}{4} = 3$$

• The slope of $\phi(x) = 3x + 1$ is $3$.

Since both lines have the same slope ($3$), they are parallel.

Conclusion

These exercises highlight the key properties of linear functions, such as intercepts, slopes, and conditions for parallelism. By solving such problems, students gain a deeper understanding of how linear equations model relationships and interact graphically. Keep practicing, and you’ll master these concepts in no time!