Chapter 7.5, Problem 15WE

To determine

Draw the given equations on the same axis.

Explanation of Solution

Given Information:

Given equations are :

  $\begin{array}{l}y=-x^2\\ y+3=-x^2\\ y-4=-x^2\end{array}$

Calculation:

• Parent Function: $y=-x^2$
• Vertex = (0,0)
• Type of Graph:

Downward Parabola

• Axis of Symmetry: Passes through the vertex and divides the parabola into two congruent halves.

  $x=0$

• In the equation $y+3=-x^2$ ; comparing it with standard form of parabola equation $Y=-X^2$
• $y+3=Y;x^2=X^2$

• Vertex :

  $\begin{array}{l}(X,Y)=(0,0)\\ \Rightarrow y+3=Y=0;x=X=0\\ \Rightarrow y=-3;x=0\\ \Rightarrow Vertex\equiv (0,-3)\end{array}$

• Find Axis of Symmetry :

  $\begin{array}{l}X=\mathrm{0........................}{\text{[axis of symmetry of Y=-X}}^2]\\ \Rightarrow x=\mathrm{0........}[X=x]\end{array}$

The equation of axis of symmetry is x = 0

• In the equation $y-4=-x^2$ ; comparing it with standard form of parabola equation $Y=-X^2$
• $y-4=Y;x^2=X^2$

• Vertex :

  $\begin{array}{l}(X,Y)=(0,0)\\ \Rightarrow y-4=Y=0;x=X=0\\ \Rightarrow y=4;x=0\\ \Rightarrow Vertex\equiv (0,4)\end{array}$

• Find Axis of Symmetry :

  $\begin{array}{l}X=\mathrm{0........................}{\text{[axis of symmetry of Y=-X}}^2]\\ \Rightarrow x=\mathrm{0........}[X=x]\end{array}$

The equation of axis of symmetry is x = 0

Use the above information to graph the equations

  $\begin{array}{l}y=-x^2\\ y+3=-x^2\\ y-4=-x^2\end{array}$: