Параболи: вершина та вісь симетрії

(Video) Parabolas

by PatrickJMT

Here's a video by patrickJMT showing you how to find the vertex and axis of symmetry of a parabola.

Summary

If a parabola has the form: $y=a(x-h)^2+k$, then you will know that:

• It has a vertex at $(h,k)$
• Opens upwards if $a>0$
• Opens downwards if $a<0$

Example

Find the vertex and direction of opening of $y=-3(x+4)^2+5$.

Try this problem on your own, then watch the video to see the solution.

Axis of Symmetry

One important thing to notice about parabolas is that they are symmetrical.

When a shape is symmetrical, one side is a mirror image of the other side.

If you draw a straight line down the middle of a parabola, the left and right sides will match. They reflect each other across that line.

This middle line is called the axis of symmetry.

In the above graph of $y=x^2$, we can see that the axis of symmetry is at $x=0$, as that is where the parabola is mirrored. This is how we can graphically determine what the axis of symmetry is.

However, we can also algebraically figure out the axis of symmetry by using the following formula:

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

This formula only works if the quadratic equation is in standard form.

Say we have the quadratic equation, $y=2x^2+2x-3$. If we want to find the axis of symmetry, we can plug in $a=2$ and $b=2$.

$$\begin{align} x &= \frac{-b}{2a} \\ x &= \frac{-2}{2 \cdot 2} \\ x &= \frac{-2}{4} \\ x &= \frac{-1}{2} \\ \end{align}$$

So, for the quadratic equation, $y=2x^2+2x-3$, the axis of symmetry is $x = \frac{-1}{2}$.

Vertex

The vertex is the point where the axis of symmetry meets the parabola.

It is the point where the quadratic function changes direction. The graph stops going down and starts going up, or stops going up and starts going down.

If the parabola opens upward (like a “U”), the vertex is the minimum point. This means it is the lowest point on the graph.

If the parabola opens downward, the vertex would be the maximum point.

In this graph, the vertex is at the minimum, $(0,0)$.

In this next graph, the quadratic equation is upside-down, so the vertex is at the maximum.

We know the formula to find the axis of symmetry, and that gives us an $x$-coordinate. To find the $y$-coordinate, we can plug in the solved $x$-value into the original quadratic equation.

If we go back to our original example from the beginning of this lesson, we found that for the quadratic equation, $y=2x^2+2x-3$, the axis of symmetry is at $x=-\frac12$.

$$\begin{align} y &= 2x^2+2x-3 \\ y &= 2(-\frac12)^2+2(-\frac12)-3 \\ y &= 2(\frac14) -1 -3 \\ y &= \frac12 -1 -3 \\ y &= -\frac72 \\ \end{align}$$

So, our vertex is at the point $(-\frac12, -\frac72 )$.

Parabola: Vertex and Axis of Symmetry

Two key aspects of a parabola are the vertex and the axis of symmetry. These concepts will be helpful when solving word problems with parabolas.

The axis of symmetry is a vertical line that divides a parabola into two equal halves. This line acts like a mirror. The left side of the parabola reflects onto the right side across this line.

The vertex is the point where the parabola meets the axis of symmetry.

The vertex can be the highest point or the lowest point on the graph. It depends on whether the parabola opens up or down.

In the next lesson, we will learn about the focus and directrix.