Question 727922
That tangent's slope is zero when
{{{2x-3=0}}} --> {{{2x=3}}} --> {{{x=3/2}}}
For {{{x<3/2}}} the tangent's slope is negative because {{{2x-3<0}}}
and for {{{x>3/2}}} the tangent's slope is positive because {{{2x-3>0}}} .
That means that the function decreases for {{{x<3/2}}},
reaches its minimum value (minimum height of the graph) at {{{highlight(x=3/2)}}}
and increases for {{{x>3/2}}}.
There are infinite quadratic functions {{{y=x^2-3x+k}}} whose tangent's slope is {{{2x-3}}} , but knowing that the graph passes through (0,4) we know that for {{{x=0}}} {{y=k=4}}} so the function is
{{{highlight(y=x^2-3x+4)}}}
However, I do not know how you would know to start with {{{y=x^2-3x+k}}}
because I never had to teach beginning calculus.