To determine the equation of the quadratic function $g$ shown in the graph and find its intercepts with the coordinate axes, let's analyze the key points and features of the parabola.

Step 1: Identify the Form of the Function

A quadratic function can generally be written as: $g(x) = ax^2 + bx + c$ Alternatively, if we have the vertex and another point, we can use the vertex form: $g(x) = a(x - h)^2 + k$ where $(h, k)$ is the vertex.

Step 2: Find the Vertex

From the graph:

• The vertex appears to be at $(-3, -4)$.
  • This suggests that the equation can take the form $g(x) = a(x + 3)^2 - 4$.

Step 3: Identify Another Point on the Curve

Another clearly marked point is the y-intercept at $(0, 2)$. Substituting this point into the equation will help us find $a$: $2 = a(0 + 3)^2 - 4$ $2 = 9a - 4$ $9a = 6$ $a = \frac{2}{3}$

Step 4: Write the Equation

Now that we have $a = \frac{2}{3}$, the equation of the function $g(x)$ is: $g(x) = \frac{2}{3}(x + 3)^2 - 4$

Step 5: Find the Intercepts

1. Y-Intercept: We've already identified this as $(0, 2)$.
2. X-Intercepts: Set $g(x) = 0$ and solve for $x$: $0 = \frac{2}{3}(x + 3)^2 - 4$ $\frac{2}{3}(x + 3)^2 = 4$ $(x + 3)^2 = 6$ $x + 3 = \pm \sqrt{6}$ $x = -3 \pm \sqrt{6}$ So the x-intercepts are $x = -3 + \sqrt{6}$ and $x = -3 - \sqrt{6}$.

Summary

• The equation of the quadratic function is: $g(x) = \frac{2}{3}(x + 3)^2 - 4$
• The intercepts are:
  • Y-intercept at $(0, 2)$
  • X-intercepts at $\left( -3 + \sqrt{6}, 0 \right)$ and $\left( -3 - \sqrt{6}, 0 \right)$

Would you like more details or have any questions?

Related Questions:

1. How can you convert a quadratic function from vertex form to standard form?
2. What are some other methods for finding the intercepts of a quadratic function?
3. How does changing the value of $a$ affect the shape of a parabola?
4. What is the axis of symmetry for this parabola, and how can it be derived?
5. How can the quadratic formula be used to find the roots of a function in standard form?

Tip:

When you know the vertex and another point, using the vertex form of a quadratic equation is a quick and efficient way to find the function's rule.