5.3 x-intercepts and the Quadratic Formula

Building on what we did last time

(What have we looked at so far?)

SAT #1Quick Poll!

SAT #2Quick Poll!

SAT #3Quick Poll!

Parabola ReviewFor y = x2 - x - 12

What is the y-intercept? The vertex? (Complete the square to find the vertex and then graph it to test your math.)

What is the symmetric point?

What are the x-intercepts? Remember, this is where y equals what? (Another hint: factor.)

These last values are also called zeros, roots, and solutions.

Problem of the DayFor y = x2 - x – 12

y-intercept: (0, -12)

Vertex (.5, -12.25)

Symmetric point (1, -12)

x-intercepts (-3, 0) and (4, 0)

Use the calculate-minimum feature of your graphing calculator to verify the vertex.

Problem of the DayFor y = x2 - x – 12

Vertex (.5, -12.25)

One more thing, we found the vertex by completing the square and by graphing. There is a third, really easy way to do it– one that you may have used already.

Think of this equation in y = ax2 + bx + c form. Find –b/2a. That is the x-coordinate for the vertex.

Then plug that value back in to find the y-coordinate for the vertex. Let’s try it out.

2 real solutionsIn the POD, we found 2 x-intercepts. How many times,

then, will the graph cross the x-axis?

We found them by factoring. What can we use if factoring is not an easy option?

2 real solutionsIn the POD, we found 2 x-intercepts. How many times,

then, will the graph cross the x-axis?

We found them by factoring. What can we use if factoring is not an easy option?

Try the Quadratic Formula. Write it down here.

The bit underneath the radical sign is called the discriminant.

2 real solutionsTry the Quadratic Formula.

y = 3x2 + 13x + 7

x =

What is the discriminant? If it is greater than 0, we will have 2 real solutions and 2 x-intercepts. Why?

2 real solutionsTry the Quadratic Formula.

y = 3x2 + 13x + 7

The discriminant equals 85; because it’s positive, we will have 2 real roots. Graph the equation on your calculators to see.

Keep in mind, the decimal answers are not exact!

70.3,63.6

8513

6

8416913

32

7341313 2

x

2 real solutionsTry the Quadratic Formula.

y = 3x2 + 13x + 7

Once you’ve graphed it, use the calculate-zero feature to verify the x-intercepts (roots/ zeros). You should get something close to the decimal approximation above.

70.3,63.6

8513

6

8416913

32

7341313 2

x

2 real solutionsTry the Quadratic Formula.

y = 3x2 + 13x + 7

1 real solutionIf the discriminant equals 0, then we have only 1 real solution, and

1 x-intercept.

Use the quadratic formula on this:

y = x2 + 4x + 4

x =

What is the discriminant?

What is/ are the roots? Graph on your calculators to see.

1 real solution

y = x2 + 4x + 4

What do you think about that x-value?

0 real solutionsIf the discriminant is negative, we have imaginary numbers, which

means no real solutions, and no x-intercepts.

Use the quadratic formula on this:

y = x2 - 3x + 15

What is the discriminant? How many x-intercepts does the graph have? Use your calculators to see.

What is the vertex of this parabola? (Use -b/2a, and then graph and calculate the minimum to test your math.)

0 real solutionsy = x2 - 3x + 15

Again, we can essentially discount the small variation from 1.5 for the x-value.