The exercise provides two functions $f(x) = 2x - 3$ and $g(x) = \frac{2 - 3x}{4}$ and requires the following tasks:

1. Graph the curves of $f$ and $g$ on an orthonormal coordinate system with a scale of 2 cm per unit.

Before plotting, let’s analyze each function.

• For $f(x) = 2x - 3$:

  • The slope is 2, meaning the line is increasing.
  • The y-intercept is -3, so the line passes through $(0, -3)$.

• For $g(x) = \frac{2 - 3x}{4}$:

  • Simplifying this expression: $g(x) = \frac{2}{4} - \frac{3x}{4} = \frac{1}{2} - \frac{3x}{4}$.
  • This is a linear function with slope $-\frac{3}{4}$ and y-intercept $\frac{1}{2}$, meaning the line is decreasing and passes through $(0, 0.5)$.

2. Determine by calculation the coordinates of the intersection points of the two curves.

To find the points of intersection, we need to solve the equation where $f(x) = g(x)$.

So, set: $2x - 3 = \frac{1}{2} - \frac{3x}{4}$

Let's solve this equation step by step:

$2x - 3 = \frac{1}{2} - \frac{3x}{4}$ Multiply the whole equation by 4 to eliminate the fractions:

$4(2x - 3) = 4\left(\frac{1}{2} - \frac{3x}{4}\right)$

This simplifies to:

$8x - 12 = 2 - 3x$

Now, add $3x$ to both sides to combine the terms with $x$:

$8x + 3x - 12 = 2$

Simplify:

$11x - 12 = 2$

Now, add 12 to both sides:

$11x = 14$

Finally, divide by 11:

$x = \frac{14}{11}$

Substitute this value of $x$ into either $f(x)$ or $g(x)$ to find $y$.

Substitute into $f(x) = 2x - 3$:

$f\left(\frac{14}{11}\right) = 2 \times \frac{14}{11} - 3 = \frac{28}{11} - 3 = \frac{28}{11} - \frac{33}{11} = \frac{-5}{11}$

So, the point of intersection is $\left( \frac{14}{11}, \frac{-5}{11} \right)$.

Would you like further clarification on the graphing part or any other aspect of this problem?

Relative Questions:

1. How do you find the slope of a line from its equation?
2. How do the slopes of two lines affect their points of intersection?
3. What does the y-intercept of a linear function represent?
4. What is the graphical meaning of solving $f(x) = g(x)$?
5. How can you interpret the result of $x = \frac{14}{11}$ geometrically?

Tip:

When solving equations involving fractions, multiplying through by the least common denominator helps eliminate the fractions quickly.