1. The problem involves understanding how changes in the equation of a parabola $y = x^2$ affect its graph. 2. The general form of a parabola is $y = a(x-h)^2 + k$, where: - $a$ controls the width and direction (up/down) of the parabola. - $h$ shifts the parabola horizontally. - $k$ shifts the parabola vertically. 3. For the first set of equations: - $y = 4x^2$ means the parabola is narrower and steeper than $y = x^2$ because $a=4 > 1$. - $y = x^2$ is the standard parabola. - $y = \frac{1}{4}x^2$ is wider and less steep because $a=\frac{1}{4} < 1$. 4. For the second set: - $y = x^2 + 4$ shifts the parabola up by 4 units. - $y = x^2$ is centered at the origin. - $y = x^2 - 4$ shifts the parabola down by 4 units. 5. For the third set: - $y = x^2$ is the standard parabola. - $y = -x^2$ opens downward because $a = -1$. 6. For the fourth set: - $y = (x+3)^2$ shifts the parabola 3 units to the left. - $y = x^2$ is centered at the origin. - $y = (x-3)^2$ shifts the parabola 3 units to the right. 7. These transformations help us understand how the parameters $a$, $h$, and $k$ affect the shape and position of parabolas. Final answer: The graphs illustrate how changing $a$ affects width and direction, changing $k$ shifts vertically, and changing $h$ shifts horizontally in the parabola equation $y = a(x-h)^2 + k$.