The image you uploaded presents traces of a function at $x = 1$ and $y = -1$, with the function given as $f(x, y) = ax^2 + by$.

Step 1: Analyze the graph for $x = 1$

The left graph represents the trace for $x = 1$. At this point, the graph appears to be a straight line with a negative slope. This suggests a linear term involving $y$, likely related to the term $by$.

Step 2: Analyze the graph for $y = -1$

The right graph represents the trace for $y = -1$. This is a parabola opening upwards, which indicates a quadratic relationship in $x$, corresponding to the $ax^2$ term.

Determining $a$ and $b$:

• From the parabola on the right (when $y = -1$), we can deduce that the coefficient $a$ is positive because the parabola opens upwards.
• From the straight line on the left (when $x = 1$), the slope appears to be negative, so $b$ is negative.

You should now be able to determine the precise values of $a$ and $b$ by considering the specific shape and values on the graph. If you need further clarification on how to calculate $a$ and $b$, feel free to ask!

Would you like more details on the steps or any specific part of the graph?

Here are some additional related questions:

1. How do you calculate the slope of a line from a graph?
2. What would happen if $a$ were negative in the quadratic graph?
3. How do you solve for unknown coefficients in an equation given a graph?
4. How can you identify the vertex of a parabola from its graph?
5. What does the negative sign of $b$ indicate in the context of the graph?

Tip: Always check the direction of the curve and its general shape (concave up or down) to determine the sign of quadratic coefficients.