Level 2 - Graphing Quadratic Functions Using Vertex Form and Intercept Form - Graphing and Solving Quadratic Equations (Algebra 1)

This lesson will discuss how to graph a particular type of function called a quadratic function.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Functions
Quadratic expressions
Intercepts

Explore

Maximum and Minimum Points of Graphs

In the coordinate plane, the graphs of three functions and their corresponding equations are shown.

Pay close attention to the maximum/minimum point of each graph. Can their coordinates be identified just by looking at the corresponding equation?

Discussion

Parabola and Quadratic Function

All three graphs of the functions presented earlier have a very distinctive form. These curves have a specific name, which now will be properly introduced.

Equations of parabolas always contain a variable raised to the second power. This is why the functions that represent vertical parabolas are called quadratic functions.

In this lesson, only the vertex and the intercept forms will be discussed.

Discussion

Characteristics of Quadratic Functions

The inherent shape of parabolas gives rise to several characteristics that all quadratic functions have in common.

Concept

Vertex of a Parabola

Because a parabola either opens upward or downward, there is always one point that is the absolute maximum or absolute minimum of the function. This point is called the vertex.

At the vertex, the function changes from increasing to decreasing, or vice versa.

Concept

Zeros of a Parabola

A parabola can intersect the $x -$ axis at zero, one, or two points. Since the function's value at an $x$ -intercept is always $0,$ these points are called zeros, or sometimes roots.

$y -$ intercept

Because all graphs of quadratic functions extend infinitely to the left and right, they each have a $y$ -intercept somewhere along the $y$ -axis.

Pop Quiz

Identifying Characteristics of a Parabola

Consider the given parabola. Identify its zeros, line of symmetry, or $y -$ intercept.

Discussion

Vertex Form of a Quadratic Function

Discussion

Graphing a Quadratic Function in Vertex Form

Example

Using the Vertex Form to Draw a Parabola

LaShay loves playing golf.

She is trying to improve her swing by drawing the parabola that the ball will make. Using her math knowledge, she has calculated the quadratic function that corresponds to this parabola.

y = - 2 (x - \frac{3}{2})^{2} + \frac{9}{2}

Draw the parabola to help LaShay improve her swing!

Answer

Hint

Identify the vertex of the parabola. Make the graph only for the first quadrant.

Solution

To identify the vertex of the parabola, the given equation can be compared to the general form of a quadratic function written in vertex form.

General Form y = a (x - h)^{2} + k Given Function y = - 2 (x - \frac{3}{2})^{2} + \frac{9}{2}

For the given function,

h = \frac{3}{2}

and

k = \frac{9}{2} .

This means that the vertex of the parabola is

(\frac{3}{2}, \frac{9}{2}) .

Since the parabola illustrates the position of the ball, the first quadrant will only be considered.

The axis of symmetry of the parabola is the vertical line through the vertex. Therefore, in this case, the equation of the axis of symmetry is $x = \frac{3}{2} .$

The

y -

intercept will now be determined and plotted. To do so,

x = 0

will be substituted in the given equation.

y = - 2 (x - \frac{3}{2})^{2} + \frac{9}{2}

Substitute

$x = 0$

y = - 2 (0 - \frac{3}{2})^{2} + \frac{9}{2}

Evaluate right-hand side

SubTerm

Subtract term

y = - 2 (- \frac{3}{2})^{2} + \frac{9}{2}

NegBaseToPosPow

$(- a)^{2} = a^{2}$

y = - 2 (\frac{3}{2})^{2} + \frac{9}{2}

PowQuot

$(\frac{a}{b})^{m} = \frac{a ^{m}}{b ^{m}}$

y = - 2 (\frac{9}{4}) + \frac{9}{2}

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

y = - \frac{1 8}{4} + \frac{9}{2}

ReduceFrac

$\frac{a}{b} = \frac{a / 2}{b / 2}$

y = - \frac{9}{2} + \frac{9}{2}

AddFrac

Add fractions

y = 0

The parabola intercepts the

y -

axis at

(0, 0) .

Next, the

y -

intercept will be reflected across the axis of symmetry.

Note that for the given function, the value of

a

- 2 .

Therefore, the parabola opens downward. This corresponds to the loci of the points plotted in the coordinate plane. Finally, these points can be connected with a smooth curve to draw the parabola.

Example

Writing the Vertex Form of a Quadratic Function Given its Graph

Mark is studying parabolas so that he can help LaShay with her golf swing. He wants to write the vertex form of the quadratic function that corresponds to the given graph.

Help Mark find the desired equation!

Hint

Start by identifying the vertex of the parabola.

Solution

Recall the format of the vertex form.

y = a (x - h)^{2} + k

In this format, the vertex of the parabola has coordinates

(h, k) .

Therefore, to state the values of

h

and

k,

start by identifying the vertex.

The vertex is the point with coordinates

(3, - 2) .

This means that in the equation that corresponds to the given parabola,

h = 3

and

k = - 2 .

y = a (x - 3)^{2} + (- 2) ⇕ y = a (x - 3)^{2} - 2

Finally, to find the value of

a,

any point on the parabola can be used. For simplicity, the

y -

intercept will be used.

The parabola intercepts the

y -

axis at

(0, 1) .

Therefore, to find the value of

a,

x = 0

and

y = 1

will be substituted into the equation and the equation will be solved for

a .

y = a (x - 3)^{2} - 2

SubstituteII

$x = 0$ , $y = 1$

1 = a (0 - 3)^{2} - 2

Solve for

a

SubTerm

Subtract term

1 = a (- 3)^{2} - 2

NegBaseToPosPow

$(- a)^{2} = a^{2}$

1 = a (9) - 2

AddEqn

$LHS + 2 = RHS + 2$

3 = a (9)

DivEqn

$LHS / 9 = RHS / 9$

\frac{3}{9} = a

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

\frac{1}{3} = a

RearrangeEqn

Rearrange equation

a = \frac{1}{3}

It has been found that the value of

a

\frac{1}{3} .

With this information, the equation of the given parabola can be written in vertex form.

y = \frac{1}{3} (x - 3)^{2} + (- 2) ⇕ y = \frac{1}{3} (x - 3)^{2} - 2

Explore

Zeros of a Parabola

In the coordinate plane below, the graphs of three quadratic functions and their corresponding equations are shown.

Three graphs of different parabolas with their zeros identified

Pay close attention to the zeros of each graph. Can their coordinates be identified just by looking at the corresponding equation?

Discussion

Factored Form of a Quadratic Function

Pop Quiz

Identifying Vertex and Factored Forms

In the following applet, several quadratic functions are expressed in different forms. Are the quadratic functions written in vertex form, factored form, or neither?

The applet randomly generates equations in vertex form, factored form, or neither of these forms

Discussion

Graphing a Quadratic Function in Factored Form

Example

Using the Factored Form to Draw a Parabola

By now, it is not a secret that LaShay loves playing golf.

She is once again trying to improve her swing by drawing the parabola that the ball will make. One more time, she uses her math knowledge to calculate the quadratic function that corresponds to this parabola.

y = - 2 x (x - 4)

Draw the parabola to help LaShay improve her swing!

Answer

Hint

Identify the zeros of the quadratic function. If necessary, rewrite the function.

Solution

To identify the zeros of the quadratic function, the given equation can be rewritten and compared to the general form of a quadratic function written in factored form.

General Form y = a (x - p) (x - q) Given Function y = - 2 x (x - 4) ⇕ y = - 2 (x - 0) (x - 4)

For the given function,

p = 0

and

q = 4 .

This means that the zeros of the function are

0

and

4,

so the parabola intersects the

x -

axis at

(0, 0)

and

(4, 0) .

Next, the parabola's axis of symmetry can be determined.

x = \frac{p + q}{2}

SubstituteII

$p = 0$ , $q = 4$

x = \frac{0 + 4}{2}

Evaluate right-hand side

IdPropAdd

\IdPropAdd

x = \frac{4}{2}

CalcQuot

Calculate quotient

x = 2

The axis of symmetry is the vertical line with equation

x = 2 .

Since the axis of symmetry is the vertical line through the parabola's vertex, the

x -

coordinate of the vertex is

2 .

To find its

y -

coordinate,

x = 2

can be substituted in the given equation.

y = - 2 x (x - 4)

Substitute

$x = 2$

y = - 2 (2) (2 - 4)

Evaluate right-hand side

MultNegPos

$(- a) b = - a b$

y = - 4 (2 - 4)

SubTerm

Subtract term

y = - 4 (- 2)

MultNegNegOnePar

$- a (- b) = a \cdot b$

y = 8

The vertex of the parabola lies at

(2, 8) .

Note that, for the given function, the value of $a$ is $- 2 .$ Therefore, the parabola opens downward. This corresponds to the loci of the points plotted in the coordinate plane. Finally, these points can be connected with a smooth curve to draw the parabola.

Example

Writing the Factored Form of a Quadratic Function Given its Graph

Continuing his studies, Mark wants to write the factored form of the quadratic function that corresponds to the given parabola.

Help Mark find the desired equation!

Hint

Start by identifying the $x -$ intercepts of the parabola.

Solution

Recall the format of the factored form of a parabola.

y = a (x - p) (x - q)

In this format, the zeros of the function are

p

and

q .

Therefore, to state the values of

p

and

q,

start by identifying the

x -

intercepts of the graph.

The parabola intersects the

x -

axis at

(- 5, 0)

and

(1, 0) .

This means that

p

and

q

are

- 5

and

1 .

It does not matter which value is attributed to which variable, so it will arbitrarily be considered that

p = - 5

and

q = 1 .

With this information, the following equation can be written.

y = a (x - (- 5)) (x - 1) ⇕ y = a (x + 5) (x - 1)

Finally, to find the value of

a,

any point on the parabola can be used. For simplicity, the vertex will be used.

The parabola's vertex lies at

(- 2, - 3) .

Therefore,

- 2

and

- 3

can be substituted for

x

and

y,

respectively, in the partial equation. The equation can then be solved for

a .

y = a (x + 5) (x - 1)

SubstituteII

$x = - 2$ , $y = - 3$

- 3 = a (- 2 + 5) (- 2 - 1)

Solve for

a

AddSubTerms

Add and subtract terms

- 3 = a (3) (- 3)

MultPosNeg

$a (- b) = - a \cdot b$

- 3 = a (- 9)

DivEqn

$LHS / (- 9) = RHS / (- 9)$

\frac{- 3}{- 9} = a

DivNegNeg

$\frac{- a}{- b} = \frac{a}{b}$

\frac{3}{9} = a

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

\frac{1}{3} = a

RearrangeEqn

Rearrange equation

a = \frac{1}{3}

It has been found that the value of

a

\frac{1}{3} .

With this information, the equation of the given parabola can be written in factored form.

y = \frac{1}{3} (x - (- 5)) (x - 1) ⇕ y = \frac{1}{3} (x + 5) (x - 1)

Pop Quiz

Identifying Vertex or Zeros From an Equation

For the following quadratic functions, identify the vertex or the zeros.

Closure

Standard Form of a Quadratic Function

The same quadratic function can be expressed in both vertex form and factored form. For example, consider the following equations.

Vertex Form : Factored Form : y = - 2 (x - 2)^{2} + 2 y = - 2 (x - 1) (x - 3)

The equations above represent the same function. To prove this, both of them will be expanded.

- 2 (x - 2)^{2} + 2 = ? - 2 (x - 1) (x - 3)

Simplify

ExpandNegPerfectSquare

$(a - b)^{2} = a^{2} - 2 a b + b^{2}$

- 2 (x^{2} - 2 x (2) + 2^{2}) + 2 = ? - 2 (x - 1) (x - 3)

CommutativePropMult

\CommutativePropMult

- 2 (x^{2} - 2 (2) x + 2^{2}) + 2 = ? - 2 (x - 1) (x - 3)

Multiply

- 2 (x^{2} - 4 x + 2^{2}) + 2 = ? - 2 (x - 1) (x - 3)

CalcPow

Calculate power

- 2 (x^{2} - 4 x + 4) + 2 = ? - 2 (x - 1) (x - 3)

Distr

Distribute $- 2$

- 2 x^{2} + 8 x - 8 + 2 = ? (- 2 x + 2) (x - 3)

Distr

Distribute $(x - 3)$

- 2 x^{2} + 8 x - 8 + 2 = ? - 2 x (x - 3) + 2 (x - 3)

Distr

Distribute $- 2 x & 2$

- 2 x^{2} + 8 x - 8 + 2 = ? - 2 x^{2} + 6 x + 2 x - 6

AddTerms

Add terms

- 2 x^{2} + 8 x - 6 = - 2 x^{2} + 8 x - 6 ✓

As shown, when expanded both the vertex form and the factored form result in the following equation.

y = - 2 x^{2} + 8 x - 6

This form is called the standard form of a quadratic function. For more information, read about standard form of a quadratic function.

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Catch-Up and Review

y-intercept

Answer

Hint

Solution

Hint

Solution

Answer

Hint

Solution

Hint

Solution

$y -$ intercept