How do you graph the quadratic function and identify the vertex and axis of symmetry for #y=1/2x^2+4x+5#?

Question

Isaiah Anthony · Accepted Answer

#x_("intercpts")=-4+-sqrt(6)larr" Exact value"#
#y_("intercept")=5#
Vertex #->(x,y)=(-4,-3)#

#color(blue)("The different way to find "x_("vertex"))#

Write as: #y=1/2(x^2+8x)+5#

Axis of symmetry #->x_("vertex")=(-1/2)xx8= -4#

#y_("vertex")=1/2(-4)^2+4(-4)+5 = -3#

Vertex#->(x,y)=(-4,-3)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("The formula method for x-intercepts")#

Given: #y=1/2x^2+4x+5#

#y=ax^2+bx+c => x=(-b+-sqrt(b^2-4ac))/(2a)#

This really is worth committing to memory.

Where #a=1/2"; "b=4"; "c=5#

#x=(-4+-sqrt(4^2-4(1/2)(5)))/(2(1/2))#

#x=-4+-sqrt(6)larr" Exact value"#

#x~~-1.55 and -6.45# to 2 decimal places
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine the vertex")#

#x_("vertex")# is mid point of intercepts which is:

#((-4-sqrt(6))+(-4+sqrt(6)))/2 =-4#

by substitution #y_("vertex")=-3#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("Determine the y-intercept")#

It is the value of the constant #c=5#

Genesis Allen · Answer

undefined To graph the quadratic function y = 1/2x^2 + 4x + 5 and identify the vertex and axis of symmetry:

1. Find the vertex using the formula: 
   Vertex x-coordinate = -b / (2a)
   Vertex y-coordinate = f(vertex x-coordinate)
   
2. Calculate the axis of symmetry: 
   Axis of symmetry equation: x = vertex x-coordinate

3. Plot the vertex on the graph.

4. Use additional points to plot the graph or determine the direction of the parabola based on the coefficient of x^2 (a).

5. Draw the axis of symmetry line on the graph.

6. Plot additional points symmetrically on both sides of the axis of symmetry if needed to complete the graph.

7. Label the vertex and axis of symmetry on the graph.

Avery Alexander · Answer

undefined To graph the quadratic function $ y = \frac{1}{2}x^2 + 4x + 5 $, follow these steps:

1. Identify the coefficients: $ a = \frac{1}{2} $, $ b = 4 $, and $ c = 5 $.
2. Use the formula for the x-coordinate of the vertex: $ x_v = -\frac{b}{2a} $.
3. Plug the value of $ x_v $ into the function to find the y-coordinate of the vertex.
4. Plot the vertex on the coordinate plane.
5. Use the axis of symmetry formula: $ x = x_v $ to draw a vertical line through the vertex.
6. Choose additional x-values on either side of the vertex and plug them into the function to find corresponding y-values.
7. Plot these points on the graph.
8. Draw a smooth curve that passes through the plotted points to represent the quadratic function.

For the given function, $ y = \frac{1}{2}x^2 + 4x + 5 $:
- The coefficient $ a = \frac{1}{2} $, $ b = 4 $, and $ c = 5 $.
- Using the formula for the x-coordinate of the vertex: $ x_v = -\frac{b}{2a} = -\frac{4}{2(\frac{1}{2})} = -4 $.
- Plug $ x_v = -4 $ into the function to find the y-coordinate of the vertex: $ y_v = \frac{1}{2}(-4)^2 + 4(-4) + 5 = 3 $.
- So, the vertex is at (-4, 3).
- The axis of symmetry is the vertical line $ x = -4 $.
- Choose additional x-values, such as -5, -3, -2, -1, 0, 1, 2, 3, and 4, and plug them into the function to find corresponding y-values.
- Plot these points on the graph.
- Draw a smooth curve passing through the plotted points to represent the quadratic function.

The vertex of the graph is (-4, 3), and the axis of symmetry is the vertical line $ x = -4 $.