Question

What is the meaning of each of the coefficients in the function definition `f(x) = x^2+2x-3` ?

Answer 1

See explanation...

Explanation:

What I think you may have in mind is the slope intercept form of the equation of a line:

`y = mx+c`

where `y` represents the vertical displacement, `m` the slope, `x` the horizontal displacement and `c` the `y` intercept.

With quadratic functions like the given `f(x)` the identification of the meaning of each coefficient is less obvious.

Given:

`f(x) = x^2+2x-3`

`f(x)` says that this is a function called `f` depending on the variable `x`. So `f(x)` is the `y` displacement.

The `-3` means that the `y` intercept is `(0, -3)`.

What about the other two terms?

To get a more understandable form of this function, we can complete the square to get it into vertex form:

`f(x) = x^2+2x-3`

`color(white)(f(x)) = x^2+2x+1-4`

`color(white)(f(x)) = (x+1)^2-4`

or more strictly:

`color(white)(f(x)) = color(blue)(1)(x-(color(blue)(-1)))^2+(color(blue)(-4))`

This is now in the form:

`f(x) = a(x-h)^2+k`

with `a=1` and `(h, k) = (-1, -4)`

The graph of this function is a sort of 'U' shape called a parabola, with `a=1` being a multiplier determining the steepness of the 'U' compared with the parent function `x^2` and `(h, k)=(-1, -4)` being the vertex.

It is not easy to immediately see the vertex in the original form `f(x) = x^2+2x-3`, so you can see why it is useful to transform the quadratic into vertex form.

graph{x^2+2x-3 [-10, 10, -5, 5]}

Answer 2

See explanation

Explanation:

Given: `f(x)=x^2+2x-3`

`f` is just name that is used to represent the mathematical structure of the equation as declared.

`f(x)` means that that structure is applied to `x`

So `f(x)` represents the structure `x^2+2x-3`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Consider "x^2)`

Compare to the standard form `y=ax^2+bx+c` the general shape of which is: `uu" if "a>0 and nn" if "a<0`

Set `y=f(x)=x^2+2x-3`

Then `a=1;b=2 and c=-3`

Consider `a=1`

If `|a|>1` then the graph is 'squeezed' narrower
if `0<|a|<1` then the graph is 'stretched' wider

So in this case, as `a=1`, the actual shape of `uu` is not widened or made narrower than the referenced base equation of `y=x^2`.

Note that `|a|` means the value is always positive. Even if the actual value of `a` is negative. Example `|-2|=|+2|=+2`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Consider "2x)`

The 2 from `2x` moves the graph sideways from the y-axis by the amount `(-1/2)xxb color(white)("ddd") ->color(white)("ddd")(-1/2)xx2=-1`.

So `x_("vertex")=-1`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Consider "c=-3)`

The `c=-3` changes the positioning of the graph such that it always passes through `y=c->(x,y)=(0,c)` Thus it 'lifts' or 'lowers' the whole thing. This has a bearing on the effect of the `2x`. As the graph is moved sideways it will at the same time lower or go up. This is because the curve must always pass through the point `y=3 ->(x,y)=(0,3)`