How do you find the important points to graph y=x^2-2x-3y=x22x3?

2 Answers

The important points are x-intercepts (-1, 0)(1,0) and (3, 0)(3,0), y-intercept (0, -3)(0,3), and the Vertex (1, -4)(1,4)

Explanation:

From the given equation y=x^2-2x-3y=x22x3 , set x=0x=0 then solve for the y-intercept

y=x^2-2x-3y=x22x3

y=0^2-2*0-3y=02203

y=-3y=3

From the given equation y=x^2-2x-3y=x22x3 , set y=0y=0 then solve for the x-intercept

y=x^2-2x-3y=x22x3

x^2-2x-3=0x22x3=0

by factoring method

(x-3)(x+1)=0(x3)(x+1)=0

x=3x=3 and x=-1x=1 when y=0y=0

so (3, 0)(3,0) and (-1, 0)(1,0) are x-intercepts

From the given equation y=x^2-2x-3y=x22x3 ,by completing the square, find the vertex

y=x^2-2x-3y=x22x3

y=x^2-2x+1-1-3y=x22x+113

y=(x-1)^2-4y=(x1)24

y--4=(x-1)^2y4=(x1)2

the vertex (h, k)=(1, -4)(1,4)

graph{y=x^2-2x-3[-20,20, -10, 10]}

have a nice day from the Philippines

Feb 8, 2016

Axis of symmetry: x=1x=1
Vertex: (1,-4)(1,4)
X-intercepts:(-1,0)(1,0) and (3,0)(3,0)

Explanation:

y=x^2-2x-3y=x22x3 is a quadratic equation in standard form, ax+bx+cax+bx+c, where a=1, b=-2, c=-3a=1,b=2,c=3. The graph of a quadratic equation is a parabola.

You need the axis of symmetry, the vertex, and the x-intercepts.

Axis of Symmetry
The axis of symmetry is an imaginary line dividing the parabola into two equal halves. The formula for the axis of symmetry is x=(-b)/(2a)x=b2a.

Substitute the given values for aa and bb into the formula for the axis of symmetry.

x=(-b)/(2a)x=b2a

x=(-(-2)/(2*1))=x=(221)=

x=2/2=x=22=

color(green)(x=1)x=1

This is the axis of symmetry, and it is also the xx value for the vertex.

Vertex
The vertex is the maximum or minimum point of a parabola. Since a>0a>0, the vertex is the minimum and the parabola opens upward.

Substitute 11 for xx into the quadratic equation and solve for yy.

y=x^2-2x-3y=x22x3

y=1^2-(2*1)-3=y=12(21)3=

y=1-2-3y=123

color(purple)(y=-4)y=4

The vertex is color(green)((1,color(purple)(-4))(1,4).

X-Intercepts
The x-intercepts are the values of xx that intersect the y-axis. A parabola has two x-intercepts.

Substitute 00 for yy in the quadratic equation.

0=x^2-2x-30=x22x3

Factor x^2-2x-3x22x3

Find two numbers that when added equal -22, and when multiplied equal -33. The numbers 11 and -33 fit the pattern. Rewrite the equation as its factors.

color(red)((x+1))color(blue)((x-3))=0(x+1)(x3)=0

First solve color(red)((x+1))=0(x+1)=0

Subtract color(red)11 from both sides.

color(red)(x=-1)x=1

Next solve color(blue)((x-3))=0(x3)=0

Add color(blue)(3)3 to both sides.

color(blue)(x=3)x=3

The x-intercepts are (-1,0)(1,0) and (3,0)(3,0).

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