A quadratic equation graph is of a general shape. In this question 7x^2 is negative. So the general shape is nn
Suppose that 7x^2 had been positive then its general shape is uu
Why is this important? It is a direct indicator of if the vertex is a maximum or a minimum.
color(brown)("Consider what I will call the 'base line graph' of "y=x^2)
color(blue)("Point 1: ") In this case the coefficient of x is (+1) the
" " general shape is uu thus the vertex in a minimum
color(blue)("Point 2: ") Suppose I add a constant of say 3. What this does is lift
" "the graph of y=x^2 upwards by 3. In the same way, if
" " the constant was -3 then it would lower the graph
color(blue)("Point 3: ")Suppose the equation was y=x^2+6x
" "The 6x shifts the graph sideways in such a way that the
" "axis of symmetry is (-1/2xx6)
color(blue)("Point 4: ")Suppose the equation was y=3x^2+7x-5
" "it would be prudent to write it as y=3(x^2+7/3x)-5
" "In this case the x_("vertex")=(-1/2)xx(7/3) = -7/6
" "Also notice that the y_("intercept") is the same as the
" "constant of -5.
color(blue)("Point 5: ")Suppose we changed the coefficient x such that we
" "hade 3x^2. Perhaps the best way of describing its
" "effect is to say that it makes the curve narrower. In the
" "same, if the coefficient of x^2 is less than 1 it makes " "the graph wider.
color(blue)("Point 6: ")To mathematically determine the x intercept set y=0 and
" "manipulate the equation. To determine the y intercept " "set x=0 and manipulate the equation.
color(brown)("Compare Point 4 with the graph below")
Tony B