Question

How do you graph the parabola `y=-3x^2+x+1` using vertex, intercepts and additional points?

Answer 1

Knowing the max/min (vertex and direction) and the intercepts, a rough parabolic shape can be drawn connecting them.

Explanation:

The intercepts (roots of the equation) and vertex (maximum or minimum point) can be calculated directly to facilitate scaling a graph. That is the extent of the conceptual sketch from the characteristic points.

The vertex is easily found by taking the derivatives of the function. The point where the first derivative is zero (0) is the vertex. If the second derivative is negative it is a maximum. If it is positive the vertex is a minimum.

The "additional points" is just a smoothing of the curve, and can may contain as many points as you want to calculate. To generate a useful list set up an x:y chart. Starting from the vertex pick an x value and calculate the y value. Continue in the same x direction until you have enough points to shape the curve.

The mirror side of the parabola is the set of points equidistant from the vertex, so you may not need to actually calculate the all.

Answer 2

Refer to the explanation for the process.

Explanation:

Note: This is a long answer.

Graph:

`y=-3x^2+x+1` is a quadratic equation in standard form:

`y=ax^2+bx+c`,

where:

`a=-3`, `b=1`, and `c=1`

Vertex: minimum or maximum point of a parabola

The axis of symmetry is the `x`-value of the vertex.

`x=(-b)/(2a)`

Plug in the known values.

`x=(-1)/(2*-3)`

Simplify.

`x=(-1)/(-6)`

`x=1/6` `larr` x-value of vertex

The `y`-value of the vertex is determined by substituting `1/6` for `x` into the equation and solve for `y`.

`y=-3(1/6)^2+1(1/6)+1`

Simplify.

`y=-3(1/36)+1/16+1`

Simplify.

`color(red)cancel(color(black)(-3))^(-1)(1/color(red)cancel(color(black)(36))^12)`

`y=-1/12+1/16+1`

The denominators must all be the same in order to add the fractions. Remember: any whole number, `n`, is `n/1`, so `1/1`.

The least common denominator is `48`. Each fraction must be multiplied by a multiplier that equals `1`. For example: `5/5=1`.

`y=-1/12xxcolor(magenta)(4)/color(magenta)(4)+1/16xxcolor(teal)3/color(teal)3+1/1xxcolor(blue)(48)/color(blue)(48`

Simplify.

`y=-4/48+3/48+48/48`

`y=47/48`

The vertex is: `(1/6,47/48)`

approximate vertex: `(0.167,0.979)`

Since `a<0`, the vertex is the maximum point and the parabola opens downward.

X-Intercepts: values of `x` when `y=0`.

Substitute `0` for `y` and solve for `x` using the quadratic formula.

`0=-3x^2+x+1`

`x=(-b+-sqrt(b^2-4ac))/(2a)`

Plug in the known values.

`x=(-1+-sqrt(1^2-4*-3*1))/(2(-3)`

`x=(-1+-sqrt(1+12))/(-6)`

Simplify.

`x=-(-1+-sqrt13)/6`

Solutions for `x`.

`x=-(-1+sqrt13)/6,` `-(-1-sqrt13)/6`

x-intercepts: `(-(-1+sqrt13)/6,0),` `(-(-1-sqrt13)/6,0)`

approximate values: `(-0.434,0),` `(0.768,0)`

Y-intercept: value of `y` when `x=0`.

Substitute `0` for `x` and solve for `y`.

`y=-3(0)^6+0+1`

`y=1`

y-intercept: `(0,1)`

Additional point.

Substitute `3` for `x` and solve for `y`.

`y=-3(3)^2+3+1`

Simplify.

`y=-3(9)+4`

`y=-27+4`

#y=-23

Point: `(3,-23)`

SUMMARY

Vertex: `(1/6,47/48)`

approximate vertex: `(0.167,0.979)`

X-intercepts: `(-(-1+sqrt13)/6,0),` `(-(-1-sqrt13)/6,0)`**

approximate values: `(-0.434,0),` `(0.768,0)`

Y-intercept: `(0,1)`

Additional Point: `(3,-23)`

Plot the points and sketch a parabola through the points. Do not connect the dots.

graph{y=-3x^2+x+1 [-12.66, 12.65, -6.33, 6.33]}