Exercise 25 Page 206

We want to draw the graph of a quadratic function written in standard form. To do so, we will follow five steps.

1. Identify $a,$ $b,$ and $c.$
2. Calculate and sketch the axis of symmetry.
3. Find and plot the vertex.
4. Find and plot the $y\text{-}$intercept and its symmetric point across the axis of symmetry.
5. Draw a smooth curve through the three plotted points.

Let's do it!

Identify $a,$ $b,$ and $c$

We will start by identifying the values of $a,$ $b,$ and $c.$ We have identified that $a=\text{-}4,$ $b=\text{-}24,$ and $c=\text{-}36.$

Axis of Symmetry

The axis of symmetry is the vertical line that divides the parabola into two mirror images. Its equation follows a specific formula. Let's substitute our given values $a=\text{-}4$ and $b=\text{-}24$ into this equation. The axis of symmetry is the line $x=\text{-}3.$

Vertex

To find the vertex of the parabola, we will need to think of $y$ as a function of $x,$ $y=f(x).$ We can write the expression for the vertex by stating the $x\text{-}$ and $y\text{-}$coordinates in terms of $a$ and $b.$ When determining the axis of symmetry, we found that $\text{-}\frac{b}{2a}=\text{-}3.$ Therefore, the $x\text{-}$coordinate of the vertex is $\text{-}3$ and the $y\text{-}$coordinate is $f(\text{-}3).$ To find this value, substitute our $x\text{-}$coordinate for $x$ in the given equation. The vertex of the parabola is $(\text{-}3,0).$

$y\text{-}$Intercept and Symmetric Point

Since in our equation we have that $c=\text{-}36,$ the $y\text{-}$intercept is $\text{-}36.$ Let's plot this point and the point symmetric across the axis of symmetry.

Graph

Since $a=\text{-}4,$ which is less than zero, we know that our parabola opens downwards. Let's draw a smooth curve connecting the three points we have. We should not use a straight edge for this!