Exercise 26 Page 607

The exercise ask us to find information about the parameters of a quadratic function in standard form. f(x) = ax^2+bx+c Let's start by reviewing the effects that the parameters a, b, and c have on the graph of a quadratic function. Then, we will analyze the graph given in the exercise to get information about its parameters.

Effects of the Parameter a

Since a multiplies the x^2-term, it can stretch or compress the parabola vertically.

• If a>1 the graph grows faster and, consequently, it becomes thinner and is stretched vertically.
• If 0vertically compressed or shrunk.

You can explore this below where the graph for the simplest quadratic function, y=ax^2, is graphed for reference. Give it a try!

If the value is negative, the same behavior holds but the graph would be upside down — the parabola gets reflected across the x-axis.

Relation of a and b

These parameters give information about the axis of symmetry (AOS). The AOS is the line x= - b2a. Since the AOS passes through the vertex, the x-coordinate of the vertex is = - b2a. The y-coordinate can be found by evaluating the quadratic function at that x-value.

Effects of the Parameter c

Note that when x= 0, the function takes the form f( 0)= c. Therefore, c is the y-intercept of the quadratic function.

Analyzing the Graph

The exercise gives us the graph shown below.

After a careful inspection we can conclude the following.

• Since the parabola opens downwards, and it is thinner than the parabola for y=x^2, we know that a should be negative and less than -1.
• As the parabola intersects the y-axis at 1, we know that c=1.
• Because the y-axis is the AOS of the parabola, we know that its equation is 0 = - b2a. This is only possible if b=0.

Putting this all together, we know that the function for the exercise's graph must be of the form shown below. y = ax^2 + 1, with a<- 1