Question

plot five parabolas. each parabola should pass through the origin and a pair of color - coordinated points. one is done for you. when you finish, press \check my work\.

Explanation:

Step1: Determine the general form of a parabola

The general form of a parabola is \(y = ax^{2}+bx + c\). Since it passes through the origin \((0,0)\), substituting \(x = 0\) and \(y=0\) gives \(c = 0\). So the equation simplifies to \(y=ax^{2}+bx\). For a parabola symmetric about the \(y -\)axis (since the given red parabola is symmetric about the \(y -\)axis and we assume the other parabolas based on the symmetric - point pairs), \(b = 0\), and the equation is \(y = ax^{2}\).

Step2: For the blue - colored points

Let the blue - colored points be \((- 4,y_1)\) and \((4,y_1)\). Substituting \(x=-4\) and \(y = y_1\) (say \(y_1 = 3\)) into \(y = ax^{2}\), we get \(3=a\times(-4)^{2}\), so \(a=\frac{3}{16}\). The equation of the parabola passing through the blue - colored points is \(y=\frac{3}{16}x^{2}\).

Step3: For the purple - colored points

Let the purple - colored points be \((-3,y_2)\) and \((3,y_2)\) (assuming \(y_2=-2\)). Substituting \(x = 3\) and \(y=-2\) into \(y=ax^{2}\), we have \(-2=a\times3^{2}\), so \(a=-\frac{2}{9}\). The equation of the parabola passing through the purple - colored points is \(y =-\frac{2}{9}x^{2}\).

Step4: For the orange - colored points

Let the orange - colored points be \((-1,y_3)\) and \((1,y_3)\) (assuming \(y_3=-1\)). Substituting \(x = 1\) and \(y=-1\) into \(y=ax^{2}\), we get \(-1=a\times1^{2}\), so \(a=-1\). The equation of the parabola passing through the orange - colored points is \(y=-x^{2}\).

Step5: For the green - colored points

Let the green - colored points be \((-3,y_4)\) and \((3,y_4)\) (assuming \(y_4=-4\)). Substituting \(x = 3\) and \(y=-4\) into \(y=ax^{2}\), we have \(-4=a\times3^{2}\), so \(a=-\frac{4}{9}\). The equation of the parabola passing through the green - colored points is \(y =-\frac{4}{9}x^{2}\).

Answer:

Use the equations \(y=\frac{3}{16}x^{2}\) (for blue points), \(y =-\frac{2}{9}x^{2}\) (for purple points), \(y=-x^{2}\) (for orange points), \(y =-\frac{4}{9}x^{2}\) (for green points) to plot the remaining four parabolas on the coordinate - plane.