Find the equation of the functions defined by the following graphs.
The x-intercepts of the function are -\dfrac{1}{2} and 2, so the solutions of the equation f\left(x\right)=0 are -\dfrac{1}{2} and 2.
We can write this fact as:
\left( x+\dfrac{1}{2} \right)\left( x-2 \right)=0
Which simplifies to:
x^{2}-\dfrac{3}{2}x-1=0
The y-intercept of the function is -2, so we multiply the last equation by 2:
2\left( x^{2}-\dfrac{3}{2}x-1 \right)=0
Which simplifies to:
2x^{2}-3x-2=0
The function matching the given graph is:
f\left(x\right)=2x^{2}-3x-2
The x-intercepts of the function are -\dfrac{5}{2} and 1, so the solutions of the equation f\left(x\right)=0 are -\dfrac{5}{2} and 1.
We can write this fact as:
\left( x+\dfrac{5}{2} \right)\left( x-1 \right)=0
Which simplifies to:
x^{2}+\dfrac{3}{2}x-\dfrac{5}{2}=0
The y-intercept of the function is -5, so we multiply the last equation by 2:
2\left( x^{2}+\dfrac{3}{2}x-\dfrac{5}{2} \right)=0
Which simplifies to:
2x^{2}+3x-5=0
The function that matches the given graph is:
f\left(x\right)=2x^{2}+3x-5
The x-intercepts of the function are -2 and 3, so the solutions of the equation f\left(x\right)=0 are -2 and 3.
We can write this fact as:
\left( x+2 \right)\left( x-3 \right)=0
Which simplifies to:
x^{2}-x-6=0
The y-intercept of the function is 12, so we multiply the last equation by -2 :
-2\left( x^{2}-x-6 \right)=0
Which simplifies to:
-2x^{2}+2x+12=0
The function that matches the given graph is:
f\left(x\right)=-2x^{2}+2x+12
The x-intercepts of the function are -4 and \dfrac{3}{2}, so the solutions of the equation f\left(x\right)=0 are -4 and \dfrac{3}{2}.
We can write this fact as:
\left( x+4 \right)\left( x-\dfrac{3}{2} \right)=0
Which simplifies to:
x^{2}+\dfrac{5}{2}x-6=0
The y-intercept of the function is -12, so we multiply the last equation by 2:
2\left( x^{2}+\dfrac{5}{2}x-6 \right)=0
Which simplifies to:
2x^{2}+5x-12=0
The function that matches the given graph is:
f\left(x\right)=2x^{2}+5x-12
The x-intercepts of the function are -1 and 1, so the solutions of the equation f\left(x\right)=0 are -1 and 1.
We can write this fact as:
\left( x+1 \right)\left( x-1\right)=0
Which simplifies to:
x^{2}-1=0
The y-intercept of the function is 3, so we multiply the last equation by -3 :
-3\left( x^{2}-1 \right)=0
Which simplifies to:
-3x^{2}+3=0
The function that matches the given graph is:
f\left(x\right)=-3x^{2}+3
The x-intercepts of the function are -2 and \dfrac{1}{2}, so the solutions of the equation f\left(x\right)=0 are -2 and \dfrac{1}{2}.
We can write this fact as:
\left( x+2 \right)\left( x-\dfrac{1}{2}\right)=0
Which simplifies to:
x^{2}+\dfrac{3}{2}x-1=0
The y-intercept of the function is -6, so we multiply the last equation by 6:
6\left( x^{2}+\dfrac{3}{2}x-1 \right)=0
Which simplifies to:
6x^{2}+9x-6=0
The function that matches the given graph is:
f\left(x\right)=6x^{2}+9x-6
The x-intercepts of the function are -1 and 3, so the solutions of the equation f\left(x\right)=0 are -1 and 3.
We can write this fact as:
\left(x+1\right)\left( x-3\right)=0
Which simplifies to:
x^{2}-2x-3=0
The y-intercept of the function is 6, so we multiply the last equation by -2 :
-2\left( x^{2}-2x-3 \right)=0
Which simplifies to:
-2x^{2}+4x+6=0
The function that matches the given graph is:
f\left(x\right)=-2x^{2}+4x+6