How do you use the important points to sketch the graph of #y = x^2 − 6x + 1#?

Question

Eli Assouline · Accepted Answer

#color(blue)("Y-int": (0,1))#
#color(red)("Vertex": (3,8))#
#color(green)("X-Intercept/Zero":(0.172,0) and (5.828,0)# The most important points should be the #color(red)"vertex"#, the #color(blue)("y-int")#, and the #color(green)"zeroes"# (if there are any). The x-coordinate of the #color(red)(vertex)# of any quadratic equation #y=ax^2+bx+c# is: #(-b)/(2a)# Plug in the x-coordinate back into the equation to find y. Do it on this equation: #b=-6# #a=1# #c=1# #(-(-6))/(2a)=3# Now you have the #color(red)(vertex)# as #(3,y)# #y=(3^2)-6*(3)+1=-8# The #color(red)(vertex)# is #(3, -8)# The #color(blue)("y-int")# of the quadratic equation #y=ax^2+bx+color(orange)(c)# is simply #color(orange)(c)#. The #color(blue)("y-int")# of this equation is #(0,1)#. To find the #color(green)"zeroes"#, plug into the quadratic formula, which is given by: #(-b+-sqrt(b^2-4ac))/(2a)# Plug in: #(6+-sqrt((-6)^2-4*1*1))/(2*1)# Simplify: #(6+-4sqrt(2))/2# #3+-2sqrt(2)# The #color(green)"zeroes"# are: #(5.828,0) and (0.172,0)#

Jace Anderson · Answer

undefined To sketch the graph of $ y = x^2 - 6x + 1 $ using important points:
1. Find the vertex using the formula $ x = \frac{-b}{2a} $.
2. Calculate the y-coordinate of the vertex by substituting the x-coordinate into the equation.
3. Find the y-intercept by setting $ x = 0 $ and solving for $ y $.
4. Find the x-intercepts by setting $ y = 0 $ and solving for $ x $.
5. Plot the vertex, y-intercept, and x-intercepts on the coordinate plane.
6. Determine the direction of the parabola (whether it opens upwards or downwards) based on the coefficient of $ x^2 $.
7. Sketch the parabola passing through the important points.