Question

How do you find the axis of symmetry, vertex and x intercepts for `y=x^2-6x+5`?

Answer 1

`"see explanation"`

Explanation:

`"given a parabola in standard form "y=ax^2+bx+c`

`"then the x-coordinate of the vertex can be found using"`

`•color(white)(x)x_(color(red)"vertex")=-b/(2a)`

`y=x^2-6x+5" is in standard form"`

`"with "a=1,b=-6,c=5`

`rArrx_(color(red)"vertex")=-(-6)/2=3`

`"substitute this value into equation for y-coordinate"`

`y_(color(red)"vertex")=3^2-6(3)+5=-4`

`rArrcolor(magenta)"vertex "=(3,-4)`

`"the axis of symmetry is vertical and passes through the"`
`"vertex with equation"`

`x=3`

`"to find x-intercepts let y = 0"`

`rArrx^2-6x+5=0`

`"the factors of + 5 which sum to - 6 are - 1 and - 5"`

`rArr(x-1)(x-5)=0`

`"equate each factor to zero and solve for x"`

`x-1=0rArrx=1`

`x-5=0rArrx=5`

`rArrx=1" and "x=5larrcolor(red)"x-intercepts"`
graph{(y-x^2+6x-5)(y-1000x+3000)=0 [-10, 10, -5, 5]}

Answer 2

The axis of symmetry is `x=3`.

The x-intercepts are `(5,0)` and `(1,0)`.

The y-intercept is `(0,5)`.

Explanation:

Given:

`y=x^2-6x+5` is a quadratic equation in standard form:

#y=ax^2+bx+c,

where:

`a=1`, `b=-6`, and `c=5`.

Axis of symmetry: vertical line that separates the parabola into two equal halves, designated `x`.

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

`x=(-(-6))/(2*1)`

`x=6/2`

`x=3`

The axis of symmetry is `x=3`.

Vertex: maximum or minimum point of the parabola, `(x,y)`. Since `a>0`, the vertex will be the minimum point and the parabola will open upward.

Substitute `3` for `x` in the equation and solve for `y`.

`y=3^2-6(3)+5=-4`

The vertex is `(3,-4)`

X-intercepts: values of `x` when `y=0`

Substitute `0` for `y`. Solve for `x`.

`0=x^2-6x+5`

Find two numbers that when added equal `6` and when multiplies equal `5`. The numbers `-5` and `-1` meet the requirements.

`0=(x-5)(x-1)`

Set each binomial equal to zero.

`(x-5)=0`

`x=5`

`(x-1)=0`

`x=1`

The x-intercepts are `(5,0)` and `(1,0)`.

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

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

`y=0^2-6(0)+5`

`y=5`

The y-intercept is `(0,5)`.

Plot the points for the vertex, x-intercepts, and y-intercept. Sketch a parabola through the points. Do not connect the dots.

graph{y=x^2-6x+5 [-14.3, 14.17, -9.97, 4.27]}