Vertex Form of Parabolas

Vertex Form of Parabolas

 $y=a\left(x-h\right)^2+k$  

3 Forms of Quadratic Functions

• Standard

• Factored

• Vertex

Fill in the Blank

 $f\left(x\right)=3x^2+8-4x$  

Find the y-intercept of the function, type ONLY the y-value:

Fill in the Blank

 $f\left(x\right)=x^2-71$  

Find the y-intercept of the function, type ONLY the y-value:

To find the x-intercepts (aka roots aka zeros)

• We can factor!

• We can use the quadratic formula!

• We can even use complete the square (but probably won't because it's going to be used elsewhere!)

Multiple Choice

Which is another way of correctly expressing

(x - 3)(x - 3)?

• (x² - 3²)

• (x² - 3)

• (x + 3)(x - 3)

• (x - 3)²

On the next slide

The next slide contains a factoring methods review video. Watch it only if you don't remember how to factor. Then proceed to the next slide (#9).

Multiple Choice

x² − 7x − 18

• (x − 9)(x + 2)

• (x + 9)(x + 2)

• (x − 9)(x - 2)

• (x − 9)(x - 2)

Multiple Choice

Factor the expression: w² - 15w - 54

• (w + 6)(w - 9)

• (w + 3)(w - 18)

• (w + 2)(w - 27)

• (w - 6)(w + 9)

Multiple Choice

Factor the expression: 10v² + 11v - 8

• (5v + 8)(2v + 1)

• (v + 8)(v - 1)

• (5v + 8)(2v - 1)

• (10v - 8)(v + 1)

Making Connections

• Ok, you know how to factor.

• Now let's factor with purpose. First, we'll factor to SOLVE equations.

• THEN, we'll factor to find the ROOTS of quadratic functions.

• I hope you'll see it's essentially the same process.

Multiple Choice

Solve.

n² - 5 = -4

• n = ±√9

• n = ±3

• n = ±1

• No real solution

Multiple Choice

What are the x-intercept/s of the function?

• (3,0)

• (-3,0)

• (9,0) & (-9,0)

• (-3, 0) & (3, 0)

Multiple Choice

What are the solutions to this equation? (x-3)(x+2)=0

• x=3,-2

• x=-3,2

• x=2,3

• x=3,2

Multiple Choice

True or false:

The solution, root, x-intercept, and zero of a problem are all the same thing.

• True

• False, because the solution and root are the same but the x-intercept and zero are different

• False, because the x-intercept and root are the same but the zero and solution are different

• False, they are all different

Multiple Choice

What are the zeros of the quadratic function?

y = x² - 2x - 15

• x = -2

  x = -15

• x = -3

  x = 5

• x = 3

  x = -5

• x = 1

  x = -16

Multiple Choice

What are the x-interepts for the graph of the function: y = x² - 3x - 10

• -5, 2

• -5, -2

• 5, -2

• 5, 2

Vertex Form

• (h, k) is the point we call the vertex

• It is the maximum or minimum point for the parabola

• You read it DIRECTLY from the equation

• Aside: Does this parabola have x-intercepts?

x-intercepts?

• No, this parabola does NOT touch the x-axis.

• Therefore, we could say that it does not have any REAL roots

• However, upon solving it, we would determine that it has IMAGINARY solutions...or roots...or zeros

• More on this next week!

• For now, watch the video on the next slide!

Converting to Vertex Form

• Complete the Square

• Remember this!

•  $\left(\frac{b}{2}\right)^2$  

Multiple Choice

Which of the following equations matches standard form of a quadratic?

• ax + by = c

• y = a(x - h)² + k

• y = ax² + bx + c

• y = mx + b

Multiple Choice

In vertex form,

f(x) = a(x - h)² + k

what does the h stand for?

• the y-coordinate of the vertex

• the x-coordinate of the vertex

• the x-intercept

• the y-intercept

Multiple Choice

The parabola y = 3(x + 1)² + 3 has a vertex of

• (3, 3)

• (1, 3)

• ( - 1, 3)

• (3, -1)

Multiple Choice

The parabola y = x² - 2x + 4 will have a vertex of

• (-1, 2)

• (1, 2)

• (-1, 3)

• (1, 3)

Multiple Choice

Convert to vertex form:

y = x² - 2x + 5

• y = (x - 1)² + 4

• y = (x - 1)² - 4

• y = (x - 1)² + 3

• y = (x - 1)² + 7

Multiple Choice

Convert to vertex form:

y = x² + 8x + 2

• y = (x + 4)² - 14

• y = (x + 4)² - 6

• y = (x + 4)² - 18

• y = (x + 4)² + 6

Multiple Choice

Convert to vertex form:

y = x² + 4x + 10

• y = (x + 4)² + 6

• y = (x + 2)² + 6

• y = (x + 4)² - 6

• y = (x + 2)² - 6

Multiple Choice

If a parabola has a vertex at (5, 0) and opens upward, how many x-intercepts will it have?

• 0

• 1

• 2

• no way to tell

Multiple Choice

If a parabola has a vertex at (-3, 7) and opens downward, how many x-intercepts will it have?

• 0

• 1

• 2

• no way to tell

Multiple Choice

Which equation matches the graph?

• $f\left(x\right)=\left(x-1\right)^2+4$

• $f\left(x\right)=\left(x+4\right)^2-1$

• $f\left(x\right)=-\left(x-1\right)^2+4$

• $f\left(x\right)=-\left(x+4\right)^2-1$

Multiple Choice

Which equations matches the graph above?

• $h\left(x\right)=\ \left(x+1\right)^2-4$

• $h\left(x\right)=-\left(x+1\right)^2-4$

• $h\left(x\right)=\left(x+4\right)^2-1$

• $h\left(x\right)=-\left(x+4\right)^2+1$

Multiple Choice

Which of the following is the correct equation for the given graph?

• f(x) = (x - 2)² - 1

• f(x) = (x + 2)² - 1

• f(x) = -(x + 2)² - 1

• f(x) = -(x - 2)² - 1

Multiple Choice

Which quadratic function is represented by the graph?

• y = 2(x - 4)² + 5

• y = 2(x + 4)² + 5

• y = -2(x - 4)² + 5

• y = -2(x + 4)² + 5

Multiple Choice

What is the equation of this graph?

• f(x) = (x -5)² + 1

• f(x) = (x - 1)² - 5

• f(x) = (x + 1)² - 5

• f(x) = (x + 5)² +1

Multiple Choice

What is the equation of this graph?

• f(x) = (x -5)² + 1

• f(x) = (x - 1)² - 5

• f(x) = (x + 1)² - 5

• f(x) = (x + 5)² +1