How would you solve/sketch the graph of g(x) = 24 + 2x - x²?

Question

Without using higher calculus. D:

Answer 1

factor 24 + 2x - x² as (6 - x)(4 + x), so you can plot x = 6 and x = -4 as x intercepts.  

line of symmetry is halfway between at (6-4)/2 = x = 1.

vertex on line of symmetry is y = (6-1)(4+1) = 25, so at (1,25).  graph opens down.

also y intercept is (0,24), and by symmetry (2,24) is also on the graph.  

Answer 2

It factorises to g(x) = (4 + x)(6 - x)

4*6 = 24 and 6 - 4 = 2

so the x intercepts are at -4 and 6

the vertex is half way between the x intercepts wheree x = 1 (so y = 25)

and the y intercept is the constant = 24

(oh and its negative so like a frown (concave down)

Answer 3

First, factor the equation.

g(x)=(6-x)(4+x)

To find x-intercepts, set g(x)=0.

0=(6-x)(4+x)

Use the zero product property.

0=6-x and 0=4+x

Solve for x

x=6 and x= -4

The coordinates for the x-intercepts are (6,0) and (-4,0) because the x-intercepts are where the curve hits the x-axis, AKA where g(x)=0.

To sketch, first draw these points on the coordinate plane. Because the x^2 term has a negative in front of it, the graph will open down like a frown (yes it rhymes), so connect the points (6,0) and (-4,0) with a downward facing parabola.

That's it, you're done!

Answer 4

The 1st thing to note about this equation is that there are no values of x that will make g not a function.  This means that the sketch of this equation is going to be smooth and curvy without any breaks.  Also, go ahead and draw your coordinate axis.  

The y intercept is at y = 24.  Go ahead and put a dot there.  

Next step:  Factor the equation into g(x) = (x - 6)(-x - 4).

From this you can see g = 0 at x = 6 and x = -4.  Go ahead an put dots on the x axis at 6 and -4.

The next thing you need to do is take the 1st derivative of g.  

g'(x) = -2x + 2

The derivative is equal to 0 at x = 1.  This means the slope of the line changes from being + to - as we cross x = 1.  Go ahead and put a vertical dashed line at x = 1.  At this point, I would go ahead and sketch the curve.  It looks like an inverted parabola with it's center about x = 1 and crossing the x axis at 4 and 6 and the y axis at 24.  

Answer 5

rewriting as g(x) = -x^2 + 2x + 24

g(x) = (-x + 6)(x + 4)

or, g(x) = -(x^2 - 2x + 24) = -(x - 6)(x + 4)

the x-intercepts will be at x = 6 and x = -4

the line of symmetry will be half-way between these at x = 1

when x = 1, y = 25==> the vertex will be at (1,25)

you have the starting points:

vertex at (1,25)

zeros at (6,0) and (-4,0)

y-intercept at (0,24), by symmetry, (2,24) will be on curve

this will be a parabola that opens downward because of the -x^2

you can find some other points to plot by letting x = 3 (and plot this y-value at x = -1 by symmetry)

Answer 6

The simplest way of all is to make an x y table. Then plot according to the values on the table.

Example: when x=0, then y=24. Then, plot on the graph (0,24) as one of the points.

This is, by the way, how you sketch any graph.