on the set of axes below, solve the following system of equations graphically and state the coordinates of…
on the set of axes below, solve the following system of equations graphically and state the coordinates of all points in the solution set. \n$y = x^2 + 4x + 4$\n$y = 2x + 7$\nyou can move the parabola by dragging the dots. graph the line by clicking twice.
Answer
Explanation:
Step1: Analyze the first equation
The equation ( y = x^2 + 4x + 4 ) can be factored as ( y=(x + 2)^2 ), which is a parabola with vertex at ((-2,0)) and opening upwards. We can find some points on the parabola: when ( x=-3 ), ( y=(-3 + 2)^2=1 ); when ( x=-1 ), ( y=(-1 + 2)^2 = 1 ); when ( x = 0 ), ( y=(0 + 2)^2=4 ).
Step2: Analyze the second equation
The equation ( y = 2x+7 ) is a linear equation in slope - intercept form (( y=mx + b )) where the slope ( m = 2 ) and the y - intercept ( b = 7 ). We can find two points on the line: when ( x = 0 ), ( y=7 ); when ( x=-3 ), ( y=2\times(-3)+7 = 1 ).
Step3: Find the intersection points
To solve the system of equations graphically, we find the points where the parabola ( y=(x + 2)^2 ) and the line ( y = 2x+7 ) intersect. We can also solve the system algebraically to verify the intersection points. Set ( (x + 2)^2=2x + 7 ). Expand the left - hand side: ( x^{2}+4x + 4=2x + 7 ). Rearrange to get a quadratic equation: ( x^{2}+4x-2x + 4 - 7=0), i.e., ( x^{2}+2x - 3=0 ). Factor the quadratic equation: ( (x + 3)(x - 1)=0 ). So ( x=-3 ) or ( x = 1 ).
- When ( x=-3 ), substitute into ( y = 2x+7 ), ( y=2\times(-3)+7=1 ).
- When ( x = 1 ), substitute into ( y = 2x+7 ), ( y=2\times1+7 = 9 ).
So the intersection points of the two graphs are ((-3,1)) and ((1,9)).