Find an equation for the ellipse. Graph the equation. center at \( (0,0) \); focus at \( (-20,0) \); vertex at \( (-25,0) \) Type the left side of the equation of the ellinse.

To find the equation of the ellipse, we need to determine the values of the semi-major axis (a) and the eccentricity (e). The center of the ellipse is at \( (0,0) \), so the equation will be in the form: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] The distance from the center to the focus is \( c \), and the distance from the center to the vertex is \( a \). The relationship between \( a \), \( b \), and \( c \) is given by: \[ c^2 = a^2 - b^2 \] Given that the focus is at \( (-20,0) \) and the vertex is at \( (-25,0) \), we can determine the values of \( c \) and \( a \): \[ c = 20 \] \[ a = 25 \] Now we can use the relationship between \( a \) and \( c \) to find \( b \): \[ c^2 = a^2 - b^2 \] \[ 20^2 = 25^2 - b^2 \] \[ 400 = 625 - b^2 \] \[ b^2 = 625 - 400 \] \[ b^2 = 225 \] \[ b = 15 \] Now we have the values of \( a \) and \( b \), we can write the equation of the ellipse: \[ \frac{x^2}{25^2} + \frac{y^2}{15^2} = 1 \] The left side of the equation of the ellipse is: \[ \frac{x^2}{625} + \frac{y^2}{225} \] To graph the equation, you can use a graphing calculator or a graphing software. The graph will be an ellipse centered at the origin with a horizontal major axis. The vertices will be at \( (-25,0) \) and \( (25,0) \), and the foci will be at \( (-20,0) \) and \( (20,0) \).