Parabola Sign Effect A425D3

1. **State the problem:** We need to sketch the graphs of the functions $y=x^2$, $y=-3x^2$, and $y=-0.25x^2$ on the same axes and understand the effect of a negative coefficient on the graph. 2. **Recall the general form:** The functions are quadratic and can be written as $y=ax^2$ where $a$ is the coefficient. 3. **Effect of $a$ on the graph:** - If $a>0$, the parabola opens upwards. - If $a<0$, the parabola opens downwards (it is reflected over the x-axis). - The absolute value of $a$ affects the "width" or steepness of the parabola: larger $|a|$ means narrower, smaller $|a|$ means wider. 4. **Analyze each function:** - $y=x^2$: $a=1$ (positive), parabola opens upwards, standard width. - $y=-3x^2$: $a=-3$ (negative), parabola opens downwards, narrower than $y=x^2$ because $|a|=3$ is larger. - $y=-0.25x^2$: $a=-0.25$ (negative), parabola opens downwards, wider than $y=x^2$ because $|a|=0.25$ is smaller. 5. **Summary:** Negative coefficients reflect the parabola over the x-axis, changing it from opening upwards to downwards. **Final answer:** A negative coefficient in $y=ax^2$ flips the parabola upside down, making it open downwards instead of upwards. The magnitude of the coefficient controls the width or steepness of the parabola.