Describing Quadratic Functions

Consider the general expression of a quadratic function, $y=a{x}^2+bx+c,$ where $a\ne 0.$ Let's note three things we can learn from this equation.

• The $y\text{-}$intercept is $c.$
• The equation of the axis of symmetry is $x=\text{-}\frac{b}{2a}.$
• The $x\text{-}$coordinate of the vertex is $\text{-}\frac{b}{2a}.$

We will start by identifying the values of $a,$ $b,$ and $c.$ $$\begin{array}{c}f(x)=2x^2-8x+5\iff f(x)=2x^2+(\text{-}8)x+5\end{array}$$ We can see that $a$ $=$ $2,$ $b$ $=$ $\text{-}8,$ and $c$ $=$ $5.$ Since the $y\text{-}$intercept is given by the value of $c,$ we know that the $y\text{-}$intercept is $5.$ Let's now substitute $a=2$ and $b=\text{-}8$ into $\text{-}\frac{b}{2a}$ to find the axis of symmetry and the $x\text{-}$coordinate of the vertex. The equation of the axis of symmetry is $x=2,$ and the $x\text{-}$coordinate of the vertex is $2.$