The problem asks to determine the equation of the quadratic function $g$, as well as the intercepts with the axes, based on the provided graph.

Step 1: Identify key points from the graph

From the image:

• The graph of the function is a parabola that opens upwards.
• The vertex of the parabola appears to be at approximately $(1, -2)$, suggesting this is a minimum point.
• The parabola intersects the y-axis at a point approximately $y = -2$.
• Another important point given on the graph is around $(4, 5)$.

Step 2: General form of a quadratic function

The general form of a quadratic function is: $g(x) = a(x - h)^2 + k$ where $(h, k)$ is the vertex of the parabola. From the graph, the vertex appears to be at $(1, -2)$, so we substitute $h = 1$ and $k = -2$ to get: $g(x) = a(x - 1)^2 - 2$

Step 3: Use another point to find $a$

We can now substitute another known point to solve for the value of $a$. From the graph, the point $(4, 5)$ lies on the curve. Substituting $x = 4$ and $y = 5$ into the equation: $5 = a(4 - 1)^2 - 2$ $5 = a(3)^2 - 2$ $5 = 9a - 2$ $7 = 9a$ $a = \frac{7}{9}$

Step 4: Write the equation of the function

Thus, the equation of the quadratic function is: $g(x) = \frac{7}{9}(x - 1)^2 - 2$

Step 5: Find the intercepts

• Y-intercept: Set $x = 0$ in the equation: $g(0) = \frac{7}{9}(0 - 1)^2 - 2 = \frac{7}{9} - 2 = -1.22$ Thus, the y-intercept is approximately $(0, -1.22)$.

• X-intercepts: Set $g(x) = 0$ and solve for $x$: $0 = \frac{7}{9}(x - 1)^2 - 2$ $\frac{7}{9}(x - 1)^2 = 2$ $(x - 1)^2 = \frac{18}{7}$ $x - 1 = \pm \sqrt{\frac{18}{7}} \approx \pm 1.6$ $x = 1 \pm 1.6$ So, the x-intercepts are approximately $x = 2.6$ and $x = -0.6$.

Final Results:

• The quadratic equation is $g(x) = \frac{7}{9}(x - 1)^2 - 2$.
• Y-intercept: $(0, -1.22)$.
• X-intercepts: approximately $(-0.6, 0)$ and $(2.6, 0)$.

Do you have any questions or would you like further details?

Related Questions:

1. What is the significance of the vertex in the graph of a quadratic function?
2. How can we use symmetry to predict the intercepts of a quadratic function?
3. Can you calculate the roots of the quadratic equation using the quadratic formula?
4. How does the value of $a$ in the equation affect the shape of the parabola?
5. What is the relationship between the axis of symmetry and the vertex of a quadratic function?

Tip:

When finding intercepts of quadratic functions, setting $y = 0$ gives the x-intercepts, while setting $x = 0$ gives the y-intercept.