Let's break down the task for the function $f(x) = \frac{2}{3}x + \frac{1}{3}$.

1. Slope and Y-Intercept

The function is in the form of $f(x) = mx + b$, where:

• $m$ is the slope.
• $b$ is the y-intercept.

From the given equation $f(x) = \frac{2}{3}x + \frac{1}{3}$:

• Slope ($m$) = $\frac{2}{3}$.
• Y-intercept ($b$) = $\frac{1}{3}$. This is the point where the graph crosses the y-axis.

2. X-Intercept

The x-intercept occurs when $f(x) = 0$, meaning we set $f(x) = 0$ and solve for $x$: $0 = \frac{2}{3}x + \frac{1}{3}$ First, subtract $\frac{1}{3}$ from both sides: $-\frac{1}{3} = \frac{2}{3}x$ Now, multiply both sides by $\frac{3}{2}$ to solve for $x$: $x = -\frac{1}{2}$ So, the x-intercept is at $x = -\frac{1}{2}$, or $\left( -\frac{1}{2}, 0 \right)$.

3. Additional Points

To add some points to the graph, let's plug in a few values of $x$.

• When $x = 0$, $f(0) = \frac{2}{3}(0) + \frac{1}{3} = \frac{1}{3}$. So, the point $(0, \frac{1}{3})$ is on the graph.
• When $x = 3$, $f(3) = \frac{2}{3}(3) + \frac{1}{3} = 2 + \frac{1}{3} = \frac{7}{3}$. So, the point $(3, \frac{7}{3})$ is on the graph.
• When $x = -3$, $f(-3) = \frac{2}{3}(-3) + \frac{1}{3} = -2 + \frac{1}{3} = -\frac{5}{3}$. So, the point $(-3, -\frac{5}{3})$ is on the graph.

Summary:

• Slope: $\frac{2}{3}$
• Y-intercept: $\left( 0, \frac{1}{3} \right)$
• X-intercept: $\left( -\frac{1}{2}, 0 \right)$
• Additional Points:
  • $(0, \frac{1}{3})$
  • $(3, \frac{7}{3})$
  • $(-3, -\frac{5}{3})$

Would you like me to help with graphing this function, or do you have any questions about these steps?

Related Questions:

1. How would the graph change if the slope were negative?
2. How can we calculate the slope if we only know two points on the line?
3. What happens to the x- and y-intercepts if we change the constant term in the equation?
4. How can we graph a linear equation using the slope and y-intercept?
5. What is the meaning of slope in real-world contexts?

Tip:

When graphing a linear function, plot the y-intercept first and use the slope to find additional points.