Video Transcript
Write the quadratic equation represented by the graph shown.
In this question, we’re given the graph of a function. And we need to determine the quadratic equation which is represented by this graph. And to do this, we’ll start by recalling there’s two ways we can represent the graph of a quadratic function. We can write this in the form 𝑦 is equal to 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 or 𝑦 is equal to 𝑎 times 𝑥 minus ℎ all squared plus 𝑘, where 𝑎, 𝑏, 𝑐, ℎ, and 𝑘 are real numbers and 𝑎 is not allowed to be equal to zero.
The first of these is called the standard form for a quadratic equation, and the second is called the vertex form. And there are positives and negatives to choosing either. However, a good rule of thumb is to check what the coordinates of the vertex of the graph is. And remember, this is the point on the curve where the line of symmetry passes through. Or alternatively, since this curve opens downwards, it will be the maximum value for the output. In this case, we can see the coordinates of this point are negative seven, zero.
And since we can determine the coordinates of the vertex of this graph, it will be easier to work with the vertex form. And this is because for a quadratic equation given in vertex form, the coordinates of the vertex will be ℎ, 𝑘. Therefore, since we found the coordinates of the vertex to be negative seven, zero, we know ℎ must be negative seven and 𝑘 must be zero. Therefore, we’ve determined the values of both ℎ and 𝑘. The only unknown left in the vertex form is the value of 𝑎.
To do this, let’s first substitute ℎ is negative seven and 𝑘 is equal to zero into the vertex form. This gives us 𝑦 is equal to 𝑎 times 𝑥 minus negative seven all squared plus zero, which we can simplify to give us 𝑦 is equal to 𝑎 times 𝑥 plus seven all squared. We need to determine the value of 𝑎. And to do this, we’re going to need the coordinates of a point which lies on the line. And when choosing this point, we should choose a point with integer coordinates, since then we can be accurate. We can find one such point with coordinates negative five, negative four.
So we substitute these coordinates into the equation for our graph. We get negative four is equal to 𝑎 times negative five plus seven all squared. And now all that’s left to do is simplify and rearrange to solve for the value of 𝑎. We have that negative five plus seven is equal to two. So we get negative four is equal to eight times two squared. And two squared is equal to four. This gives us negative four is equal to four 𝑎. We divide both sides of the equation through by four to get that 𝑎 is equal to negative one.
Finally, we just substitute 𝑎 is equal to negative one into the vertex form to find the equation of this graph. We were able to show this is a graph of the quadratic equation 𝑦 is equal to negative one times 𝑥 plus seven all squared.
