1. The problem is to understand the vertex form of a quadratic function, which is given by the formula: $$y = a(x - h)^2 + k$$ 2. In this formula, $a$ controls the width and direction of the parabola (if $a > 0$, it opens upwards; if $a < 0$, it opens downwards). 3. The point $(h, k)$ is the vertex of the parabola, which is the highest or lowest point depending on the sign of $a$. 4. This form is useful for graphing because it directly shows the vertex and how the parabola shifts from the origin. 5. For example, if $a = 1$, $h = 2$, and $k = 3$, the function is: $$y = 1(x - 2)^2 + 3$$ which means the parabola has its vertex at $(2, 3)$ and opens upwards. 6. To expand this, you can use the formula: $$y = a(x^2 - 2hx + h^2) + k = ax^2 - 2ahx + ah^2 + k$$ 7. This shows the quadratic in standard form, but the vertex form is often easier for graphing and understanding transformations.