Draw the region enclosed by the curves x = y^2 − 4 and x = 2y + 4.
Answer (2)
To find the region enclosed by the curves x = y 2 − 4 (a parabola) and x = 2 y + 4 (a straight line), identify their intersection points, which are ( 12 , 4 ) and ( 0 , − 2 ) . Sketch both graphs, shading the area between them from y = − 2 to y = 4 .
;
To draw the region enclosed by the curves x = y 2 − 4 and x = 2 y + 4 , we need to follow these steps:
Understand the Equations:
The equation x = y 2 − 4 is a parabola that opens to the right.
The equation x = 2 y + 4 is a straight line with a slope of 2.
Find the Points of Intersection:
To find where these two curves intersect, set them equal to each other: y 2 − 4 = 2 y + 4
Rearrange this to form a quadratic equation: y 2 − 2 y − 8 = 0
Factor the quadratic equation: ( y − 4 ) ( y + 2 ) = 0
Solve for y : y = 4 or y = − 2
Substitute these y values back into one of the original equations to find x :
If y = 4 : x = 2 ( 4 ) + 4 = 12
If y = − 2 : x = 2 ( − 2 ) + 4 = 0
The points of intersection are ( 12 , 4 ) and ( 0 , − 2 ) .
Sketch the Graphs:
Draw the parabola x = y 2 − 4 , which opens to the right with the vertex at ( − 4 , 0 ) .
Draw the line x = 2 y + 4 , which passes through the points ( 12 , 4 ) and ( 0 , − 2 ) .
Shade the Region Enclosed:
The enclosed region is between the parabola and the line, starting at y = − 2 and ending at y = 4 .
By sketching these steps out on a graph, you'll be able to visualize the region enclosed by the two curves.
