Question Video: Simplifying Rational Functions Using Factorization Then Finding Their Zeros Mathematics • Third Year of Preparatory School

Video Transcript

Write the quadratic equation represented by the graph shown.

We begin by observing the general shape of the given graph. It is a parabola, meaning that it is the graph of a quadratic function and it has a line of symmetry given by the 𝑦-axis or the equation 𝑥 equals zero. This means its equation will be of the form 𝑓 of 𝑥 equals 𝑘 times 𝑥 squared plus 𝑐, with 𝑘 not equal to zero.

Our job now will be to identify the values of 𝑘 and 𝑐. We recall that the 𝑦-intercept of the function is found by calculating 𝑓 of zero. For an equation of the form 𝑓 of 𝑥 equals 𝑘 times 𝑥 squared plus 𝑐, we find that 𝑓 of zero equals 𝑐. The quadratic graph shown has a 𝑦-intercept of two. Thus, 𝑐 equals two. This allows us to write our equation as 𝑓 of 𝑥 equals 𝑘 times 𝑥 squared plus two.

Next, we can calculate the value of 𝑘 by choosing any point that lies on the given parabola. Let’s choose two, negative two. This tells us that 𝑓 of two equals negative two. We will proceed by substituting two for 𝑥 in our equation. By simplifying, we show that 𝑓 of two equals four 𝑘 plus two. To solve for 𝑘, we will use the fact that 𝑓 of two also equals negative two from the graph. This gives us the equation negative two equals four 𝑘 plus two. To solve this equation, we subtract two from each side of the equation. Then, we divide by four. The result is 𝑘 equals negative one. In fact, we would expect 𝑘 to be less than zero since the parabola opens down, or is n shaped.

Now that we know that 𝑐 equals two and 𝑘 equals negative one, we can write the equation for the given quadratic graph. The quadratic equation represented by the graph shown is 𝑓 of 𝑥 equals negative 𝑥 squared plus two.