Question

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

Answer 1

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

Explanation:

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

`y=x^2-2x-3`

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

`y=-3`

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

`y=x^2-2x-3`

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

by factoring method

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

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

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

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

`y=x^2-2x-3`

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

`y=(x-1)^2-4`

`y--4=(x-1)^2`

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

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

have a nice day from the Philippines

Answer 2

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

Explanation:

`y=x^2-2x-3` is a quadratic equation in standard form, `ax+bx+c`, where `a=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)`.

Substitute the given values for `a` and `b` into the formula for the axis of symmetry.

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

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

`x=2/2=`

`color(green)(x=1)`

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

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

Substitute `1` for `x` into the quadratic equation and solve for `y`.

`y=x^2-2x-3`

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

`y=1-2-3`

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

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

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

Substitute `0` for `y` in the quadratic equation.

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

Factor `x^2-2x-3`

Find two numbers that when added equal `-2`, and when multiplied equal `-3`. The numbers `1` and `-3` fit the pattern. Rewrite the equation as its factors.

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

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

Subtract `color(red)1` from both sides.

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

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

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

`color(blue)(x=3)`

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

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