What is the axis of symmetry of the function f/x )= − x 9 )( x − 21 )? X − 15x − 6x 6x 15?

Algebra Examples

Find the Axis of Symmetry y=-x^2+6x-15

Step 1

Rewrite the equation in vertex form.

Complete the square for .

Use the form , to find the values of , , and .

Consider the vertex form of a parabola.

Find the value of using the formula .

Substitute the values of and into the formula .

Cancel the common factor of and .

Move the negative one from the denominator of .

Find the value of using the formula .

Substitute the values of , and into the formula .

Substitute the values of , , and into the vertex form .

Set equal to the new right side.

Step 2

Use the vertex form, , to determine the values of , , and .

Step 3

Since the value of is negative, the parabola opens down.

Opens Down

Step 5

Find , the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

Substitute the value of into the formula.

Cancel the common factor of and .

Move the negative in front of the fraction.

Step 6

The focus of a parabola can be found by adding to the y-coordinate if the parabola opens up or down.

Substitute the known values of , , and into the formula and simplify.

Step 7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

#f(x)=-3x^2+3x-2#

The general formula for a quadratic equation is #ax^2+bx+c#.

#a=-3#

#b=3#

The graph of a quadratic equation is a parabola. A parabola has an axis of symmetry and a vertex. The axis of symmetry is a vertical line the divides the parabola into to equal halves. The line of symmetry is determined by the equation #x=(-b)/(2a)#. The vertex is the point where the parabola crosses its axis of symmetry, and is defined as a point #(x,y)#.

Axis of Symmetry

#x=(-b)/(2a)=(-3)/(2(-3))=-3/-6=1/2#

The axis of symmetry is the line #x=1/2#

Vertex

Determine the value for #y# by substituting #y# for #f(x)# and by substituting #1/2# for #x# in the equation,

#y=-3x^2+3x-2#

#y=-3(1/2)^2+3(1/2)-2#

#y=-3(1/4)+3/2-2#

#y=-3/4+3/2-2#

The common denominator is #8#.

#y=-3/4*2/2+3/2*4/4-2*8/8# =

#y=-6/8+12/8-16/8# =

#y=-10/8#

#y=-5/4#

The vertex is #(x,y)=(1/2,-5/4)#

X-Intercept

The x-intercepts are where the parabola crosses the x-axis.There are no x-intercepts for this equation because the vertex is below the x-axis and the parabola is facing downward.

Y-Intercept

The y-intercept is where the parabola crosses the y-axis. To find the y-intercept, make #x=0#, and solve the equation for #y#.

#y=-3(0)^2+3(0)-2# =

#y=-2#

The y-intercept is #-2#.

graph{y=-3x^2+3x-2 [-14, 14.47, -13.1, 1.14]}

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]}

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