Let's analyze the graph to solve the following questions:

The graph represents a linear function $f(x)$. To determine its equation and solve the given problems, we need two points on the line.

Step 1: Identify Points on the Line

From the graph, two clear points on the line are:

1. $(-1, -2)$
2. $(1, 2)$

Step 2: Determine the Slope

The slope $m$ of a line is given by the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the points $(-1, -2)$ and $(1, 2)$: $m = \frac{2 - (-2)}{1 - (-1)} = \frac{2 + 2}{1 + 1} = \frac{4}{2} = 2$

So, the slope of the line is $2$.

Step 3: Find the Equation of the Line

We can use the point-slope form to find the equation of the line: $y - y_1 = m(x - x_1)$ Using point $(1, 2)$ and $m = 2$: $y - 2 = 2(x - 1)$ Simplifying: $y - 2 = 2x - 2 \quad \Rightarrow \quad y = 2x$ Thus, the equation of the line is: $f(x) = 2x$

Step 4: Solve the Questions

(a) $f(2)$: $f(2) = 2(2) = 4$ So, $f(2) = 4$.

(b) $f(0)$: $f(0) = 2(0) = 0$ So, $f(0) = 0$.

(c) Find the value of $x$ for which $f(x) = 2$: $2x = 2 \quad \Rightarrow \quad x = 1$

(d) Find the value of $x$ for which $f(x) = -4$: $2x = -4 \quad \Rightarrow \quad x = -2$

Final Answers:

• (a) $f(2) = 4$
• (b) $f(0) = 0$
• (c) $x = 1$ for $f(x) = 2$
• (d) $x = -2$ for $f(x) = -4$

Would you like any more details or explanations on these steps?

Here are some related questions to consider:

1. What is the y-intercept of the function $f(x) = 2x$?
2. What is the x-intercept of the function $f(x) = 2x$?
3. How does the slope of a line affect its steepness?
4. If the slope were negative, how would the line look?
5. Can you determine the equation of a line if you're only given the slope?

Tip: The slope of a line is crucial in determining how the line behaves; a larger slope means a steeper line, while a smaller slope means a shallower line.