Станадартний вигляд квадратичної функції

A quadratic equation will be seen in several different forms. These forms are the standard form of a quadratic, vertex form, and slope-intercept form. They are useful for performing different operations on a quadratic equation.

Standard form is used for performing the quadratic formula on, or finding if the equation is a polynomial function. $$y = ax^2 + bx + c$$

Vertex form is used to find the vertex of the function to graph it. $$y = a(x - h)^2 + k$$

Intercept form is used for completing the square or finding the intercepts or solutions of the graph. $$y = a(x - p)(x - q)$$

Converting: Vertex Form $\rightarrow$ Standard Form

To convert from vertex form to standard form, we want to expand the equation in vertex form, combine like terms, and rearrange the terms.

$$y = a(x - h)^2 + k \rightarrow y = ax^2 + bx + c$$

Example

Let's convert $y=2(x-3)^2 +5$ into standard form.

$$\begin{align} y&=2(x-3)^2 +5 \\ y&=2(x^2 -6x +9) +5 \\ y&= 2x^2 -12x +18 + 5 \\ y&= 2x^2 -12x +23 \\ \end{align}$$

$y= 2x^2 -12x +23$ is in standard form!

Converting: Standard Form $\rightarrow$ Vertex Form

To convert from standard form to vertex form, we will use our steps for completing the square, which we learned about in the previous lesson.

There is one more way to convert from standard form to vertex form, and that is to use the formula for the vertex. For a vertex, $(h,k)$,

$$h=-\frac{b}{2a}$$

Once we find the vertex, we'll plug in the values for the $x-$ and $y-$coordinates of the vertex into the vertex form of a quadratic.

Example

Let's convert $y=2x^2 +4x +3$ into vertex form.

$$h = \frac{-4}{4} = -1$$

We've found our $x$-coordinate (or the axis of symmetry), so now let's find the $y$-coordinate for the vertex by plugging this solved $x$-value back into the equation.

$$\begin{align} y &= 2(-1)^2 +4 (-1) +3 \\ y &= 2-4+3 \\ y &= -2 +3 \\ y &= 1 \\ \end{align}$$

So, the vertex of the equation is $(-1,1)$. From our standard form, we know that $a=2$, so we plug in these values into vertex form.

$$y=2(x+1)^2 +1$$

Converting: Intercept Form $\rightarrow$ Standard Form

Similar to what we saw earlier when we were converting from vertex form to standard form, when we convert from intercept form to standard form, we want to expand the equation, combine like terms, and rearrange the terms.

$$y = a(x - p)(x - q) \rightarrow y=ax^2+bx+c$$

Example

Let's convert $y=3(x-2)(x+4)$ into standard form.

$$\begin{align} y&= 3(x-2)(x+4) \\ y&= 3(x^2 +4x -2x -8) \\ y&= 3(x^2 +2x -8) \\ y&= 3x^2 +6x -24 \\ \end{align}$$

We can see that $y= 3x^2 +6x -24$ is in standard form!

Converting: Standard Form $\rightarrow$ Intercept Form

At first, this may seem a little confusing. Think back to Algebra 1 when we learned how to factor quadratic equations. We looked for factors of $ac$ that add to $b$. The only difference here is that we want $a=1$, so we will first factor out $a$ if $a=1$.

$$y=ax^2 +bx +c \rightarrow y = a(x - p)(x - q)$$

Example

Let's convert $y=2x^2 +8x +6$ into intercept form.

We need $a=1$, so let's first factor out a monomial, $2$.

$$\begin{align} y &=2x^2 +8x +6 \\ y &=2(x^2 +4x +3) \\ \end{align}$$

Now, we'll find the factors for the quadratic within the parentheses. We are looking for factors of $ac$, $3$, that add to $b$, $4$.

$$\begin{align} y &=2(x^2 +4x +3) \\ y &=2(x+3)(x+1) \\ \end{align}$$

We can see that $y =2(x+3)(x+1)$ is in intercept form!

Quadratics Functions

Quadratics are polynomials of degree two. They are in the form $$ax^2+bx+c$$ where $a,b,c$ are the coefficients of the quadratic.

Graphs

Forms of Quadratics

There are three forms of quadratics:

$$\begin{align} &\text{Standard Form: }ax^2+bx+c \\ \\ &\text{Vertex Form: }a\left(b(x-h)^2\right)+k \\ \\ &\text{Factored Form: }a(x-q)(x-r) \end{align}$$

We already went over standard form above.

In vertex form, $(h,k)$ is the coordinate of the vertex of the quadratic. $a$ determines the vertical stretch and compression. If $a<0$, then the quadratic turns upside down(reflected over the $x$-axis). If $|a|>1$, the quadratic is vertically stretched by a factor of $a$. If $0<|a|<1$, then the quadratic is vertically compressed by a factor of $a$.

$b$ determines the horizontal stretch and compression. If $b<0$, the quadratic is reflected over the $y$-axis. If $|b|>1$, the quadratic is horizontally compressed by a factor of $\frac{1}{b}$. If $0<|b|<1$, the quadratic is horizontally stretched by a factor of $\frac{1}{b}$.

Factored Form

Quadratics can be factored into a product of a constant, and two binomials. As an example, the quadratic in the graph above, $x^2-11x+24$ can be factored as $(x-3)(x-8)$. The $x$-intercepts, or zeroes of the quadratic are at $x=r$ and $x=q$.

Checkpoint

(Video) Changing Between Quadratic Forms

by Spiffy Math

This video by Spiffy Math discusses the pros and cons of each quadratic form, as well as how to convert between the forms.

Summary

There are three different forms of a quadratic that we can work with.

The first type is factored. The terms of the factored forms are the factors of the quadratic, or the solutions.

The second type is standard. This is the type of quadratic we are most used to seeing, where the equation looks like $ax^2 + bx + c$.

The third type is vertex. You can tell where the vertex comes from when we use vertex form. This form is the most helpful when we are looking to sketch a graph without a lot of information.

Conversions

To go from factored form to standard form, we would use FOIL. To do the opposite (i.e., to go from standard form to factored form, we would look for factors of $ac$ that added to $b$.

To go from standard form to vertex form, we want to complete the square. To convert from vertex form to standard form, we use FOIL or expand the terms.

Example

Pause the video and try this example on your own!

$$x^2 + 4x +3$$

Given this equation in standard form, convert the quadratic to factored form and to vertex form.

Converting Between Quadratic Forms

We can convert between quadratic forms, depending on what we'd need the quadratic for.

Given the quadratic equation in factored form: $$y=2(x+3)(x-4)$$ We can convert this equation into standard form and vertex form.

Factored Form $\rightarrow$ Standard Form

To convert the given factored form into the standard form $y=ax^2+bx+c$, we need to expand the expression.

$$y=2(x+3)(x-4)$$

You can use the FOIL method to expand the binomials, then simplify. We find that the equation in standard form is:

$$y=2x^2-2x-24$$

Standard Form $\rightarrow$ Vertex Form

To convert from standard form to vertex form, we need to complete the square. If we do that, we find that the equation in vertex form is:

$$y=2(x-\frac12)^2 - \frac{49}{2}$$

In Summary

• Factored Form: $y=2(x+3)(x-4)$

• Standard Form: $y=2x^2-2x-24$

• Vertex Form: $y=2(x-\frac12)^2 - \frac{49}{2}$

You can use these steps to convert any quadratic equation between these forms.