To find the vertex of the quadratic equation [tex]\( y = 2x^2 - 4x + 3 \)[/tex], we need to find the axis of symmetry and use it to determine the vertex.
### Step 1: Identify the coefficients
The given quadratic equation is [tex]\( y = 2x^2 - 4x + 3 \)[/tex]. From this equation, we can identify the coefficients:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = 3 \)[/tex]
### Step 2: Find the axis of symmetry
The formula for the axis of symmetry for a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] is given by:
[tex]\[ x = \frac{-b}{2a} \][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = \frac{-(-4)}{2(2)} \][/tex]
[tex]\[ x = \frac{4}{4} \][/tex]
[tex]\[ x = 1 \][/tex]
### Step 3: Find the y-coordinate of the vertex
Now that we have the x-coordinate of the vertex (which is 1), we need to find the corresponding y-coordinate by plugging [tex]\( x = 1 \)[/tex] back into the original equation.
Substitute [tex]\( x = 1 \)[/tex] into [tex]\( y = 2x^2 - 4x + 3 \)[/tex]:
[tex]\[ y = 2(1)^2 - 4(1) + 3 \][/tex]
[tex]\[ y = 2 - 4 + 3 \][/tex]
[tex]\[ y = 1 \][/tex]
### Step 4: Write the vertex
The vertex of the quadratic equation [tex]\( y = 2x^2 - 4x + 3 \)[/tex] is the point [tex]\((1, 1)\)[/tex].
Therefore, the vertex of the quadratic equation is:
[tex]\[ (1.0, 1.0) \][/tex]
