Question

How do you determine whether the graph of `y = −3x^2 + 10x − 1` opens up or down and whether it has a maximum or minimum point?

Answer 1

As the given equation has `-3x^2` as the first term then the graph is of form `nn`. That is, it opens down. Consequently the graph has a maximum point.

Explanation:

`color(blue)("Answering the question")`

If the coefficient of `x^2` term is negative then the graph is of general form `nn` and thus a maximum.

If the coefficient of the `x^2` term is positive then the graph is of general form `uu` and thus a minimum.

As the given equation has `-3x^2` then the graph is of form `nn` thus it has a maximum.
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`color(blue)("Usefull tip")`

Write the equation as `y=-3(x^2+(10/(-3))x) -1`

Then the `x` value at the maximum is:

`(-1/2)xx(10/(-3))=+10/6=5/3`

Answer 2

It depends on the sign of `a`, the coefficient of the `x^2` term.

Explanation:

Asking whether a parabola opens up or down, and whether it has a maximum or minimum point, is actually the same question:

It all depends on the sign of the `x^2` term which we call `a`

`y =ax^2 +bx+c`

If `a` is positive, (`" "+ax^2" "`) then the graph opens upwards, which also means that there is a minimum turning point.
(Smiling face parabola)

`a > 0 rarr "minimum TP"`

If `a` is negative, (`" "-ax^2" "`) then the graph opens downwards, which also means that there is a maximum turning point.
(Sad face parabola)

`a < 0 rarr "maximum TP"`

So, just look at the coefficient of `x^2` and you will know immediately!