How do you graph and label the vertex and axis of symmetry #y=-4/3x^2+8/3x+8/3#?

Question

Eva Adamson · Accepted Answer

The vertex form is #y=-4/3(x-1)^2+4#

We employ

#(a-b)^2=a^2-2ab+b^2#

To determine the vertex form, we factorize and finish the square.

#y = -4/3x^2 + 8/3x + 8/3#

#y=-8/3+-4/3(x^2-2x)#

#y=-8/3+4/3#+-4/3(x^2-2x+1)#

#y=-12/3 + -4/3(x-1)^2#

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

The vertex of this parabola is located at #V=(1,4)#.

#x=1# is the symmetry axis.

#=(0,8/3)# is the intercept with the y-axis.

When #y=0#, the x-axis intercepts occur.

#0=-4/3(x-1)^2+4#

#4/3(x-1)^2=4#

#(x-1)^2=3#

#x-1=+-sqrt3#

#x=1 + -sqrt3#

#(1+sqrt3,0)# and #(1-sqrt3,0)# are the intercepts.

We can now draw the graph.

graph{(y+4/3(x-1)^2-4)((x-1)^2+(y-4)^2-0.02)(y-1000(x-1))=0 [-11.25, 11.25, -5.625, 5.625]}

Hazel Anglin · Answer

undefined To graph the quadratic equation $y = -\frac{4}{3}x^2 + \frac{8}{3}x + \frac{8}{3}$, follow these steps:

1. Find the vertex using the formula: $x_v = -\frac{b}{2a}$ and $y_v = f(x_v)$.
2. Find the axis of symmetry using the equation $x = x_v$.
3. Plot the vertex and axis of symmetry on the graph.
4. Determine two additional points by choosing values for $x$ and calculating the corresponding $y$ using the equation.
5. Plot these points and draw the parabola passing through them.
6. Label the vertex, axis of symmetry, and any other relevant points on the graph.