ANSWER
[tex]y=-\frac{1}{4}x^2+\frac{3}{4}x+1[/tex]
EXPLANATION
We want to find the equation for the function graphed.
We see that the function passes through -1 and 4 on the x-axis and 1 is the y-intercept.
These are called the roots of the function. The roots are the x values where the function equals 0.
We can find the function by applying the roots:
[tex]y=a\mleft(x-x_1\mright)\mleft(x-x_2\mright)[/tex]
where x1 and x2 are the roots.
a = leading coefficient
Therefore, we have that:
[tex]\begin{gathered} y=a(x-(-1))(x-4) \\ y=a(x+1)(x-4)_{} \end{gathered}[/tex]
Expand the brackets:
[tex]\begin{gathered} y=a(x^2-4x+x-4) \\ y=ax^2-4ax+ax-4a \\ y=ax^2-3ax-4a \end{gathered}[/tex]
Now, we have to find a.
To do that, we use the y-intercept of the function (0, 1). The y-intercept is the value of the function when x is 0.
Therefore, when x = 0, y = 1:
[tex]\begin{gathered} 1=0-0-4a \\ \Rightarrow-4a=1 \\ a=-\frac{1}{4} \end{gathered}[/tex]
Now, substitute the value of a into the function obtained earlier.
[tex]\begin{gathered} y=-\frac{1}{4}x^2-3(-\frac{1}{4})x-4(-\frac{1}{4}) \\ \Rightarrow y=-\frac{1}{4}x^2+\frac{3}{4}x+1 \end{gathered}[/tex]
That is the equation of the function graphed.
