Revisão $y=ax^2+bx+c$

(Video) Horizontal and Vertical Shifts

by mathman1024

This video by mathman1024 shows how to sketch a graph given the vertex form.

$\large{\text{Summary}}$

The vertex form of a parabola is:

$$f(x)=(x - h)^2\pm k$$

Here the $h$ is the horizontal translation because it is inside the parentheses that gets squared. Subtracting a positive number moves the vertex to the right. Subtracting a negative number (the same as adding a positive) moves the vertex to the left.

The $b$ is the vertical shift because it is outside the square part. Adding a positive number shifts the vertex upwards. Adding a negative (the same as subtracting a positive) shifts the vertex downward.

The example he graphs is: $f(x)=(x-2)^2+1$. We are subtracting $2$ inside the parentheses. This a subtraction of a positive number, so there is a horizontal shift to the right $2$ units. We add $1$ outside of the parentheses. This is adding a positive, so there is a vertical shift up $1$ unit..

Sketching Graphs of Quadratic Functions

Let's make sure we know the definitions of some important terms. These will help us graph later.

Parabola: graph of quadratic function in the standard form $f(x)=ax^2+bx+c$. If $a$ is positive, the parabola opens upward, and if $a$ is negative, the parabola opens downward.

Vertex form: $f(x)=a(x-h)^2+k$

Axis of symmetry: $x=h$ is the vertical line that goes straight down the middle of the parabola.

Vertex: $(h,k)$ or the point where the axis of symmetry meets the parabola.

Minimum point: the lowest point of a parabola

Maximum point: the highest point of a parabola

Parent quadratic function: $f(x)=x^2$.

Every quadratic equation can be manipulated into the form $f(x)=a(x-h)^2+k$. The easiest way to sketch any graph is to make a table of values. You do this by plugging in values for $x$ and calculating the outputs. Then, you plot that point on the graph and continue until you see a pattern. Quadratic functions will always give you a parabola. However, the vertex form helps us graph without a table.

In the equation $f(x)=a(x-{\color{green}{h}})^2+{\color{blue}{k}}$, the vertex is $({\color{green}{h}},{\color{blue}{k}})$.

The $h$ in the equation has a negative is being subtracted. This means when you find the vertex, you take the opposite of the number. For example, if you have the equation $f(x)=a(x-{\color{green}{2}})^2+{\color{blue}{1}}$, the vertex is $(2, 1)$.

Next, you want to look at the shifts. The parent parabola is $x^2$. A shift rightward would look like $f(x)=a(x-{\color{#8925AE}{h}})^2+k$ when $h$ is positive, but there is a negative sign because it is subtracted from $x$. A shift leftward would look like $f(x)=a(x+{\color{#8925AE}{h}})^2+k$ when $h$ is negative, but there is a positive sign because it is subtracted from $x$.

A shift upward would look like $f(x)=a(x-h)^2{\color{#8925AE}{+k}}$ when $k$ is positive. A shift downward would look like $f(x)=a(x-h)^2{\color{#8925AE}{-k}}$ when $k$ is negative.

Remember, the coordinates of the point $(h, k)$ is also called the vertex. The vertical line at $x=h$ is known as the axis of symmetry, and a parabola is symmetric (both sides of the axis of symmetry are mirror images of each other).

Another way we can sketch graphs is that we can solve for the zeroes, using different factoring or solving methods, like completing the square.

Sketching Quadratic Graphs

Take a look at $a$ in a quadratic equation $y=ax^2+bx+c.$ It can tell you a lot about the parabola!