Question Video: Identifying the Rule of a Quadratic Function from Its Graph | Nagwa Question Video: Identifying the Rule of a Quadratic Function from Its Graph | Nagwa

# Question Video: Identifying the Rule of a Quadratic Function from Its Graph Mathematics • Third Year of Preparatory School

## Join Nagwa Classes

Which of the following is the equation of the function drawn on the graph? [A] π(π₯) = βπ₯Β² + 8 [B] π(π₯) = π₯Β² β 8 [C] π(π₯) = βπ₯Β² β 8 [D] π(π₯) = π₯Β² + 8 [E] π(π₯) = π₯ β 8

04:15

### Video Transcript

Which of the following is the equation of the function drawn on the graph? (a) π of π₯ equals negative π₯ squared plus eight. (b) π of π₯ equals π₯ squared minus eight. (c) π of π₯ equals negative π₯ squared minus eight. (d) π of π₯ equals π₯ squared plus eight. Or (e) π of π₯ equals π₯ minus eight.

Letβs begin by identifying some key features of the function drawn on the graph. First, we observe that it is a parabola, and in fact it is a positive or U-shaped parabola which opens upwards. This tells us that the function drawn on the graph is a quadratic function. All parabolas are symmetric with a vertical line of symmetry through their vertex. And we observe that the line of symmetry for this function is the line π₯ equals zero or the π¦-axis. Together These two pieces of information tell us that the function drawn on the graph has an equation of the form π of π₯ equals ππ₯ squared plus π. And what we need to do is determine the values of the constants π and π.

Now the value of π is the π¦-intercept of the graph. We know this because the π¦-intercept occurs when π₯ is equal to zero. And if we are to evaluate the function π of zero, we obtain π multiplied by zero squared plus π, which is simply π. From the graph, we can see that the π¦-intercept of this function is negative eight, so the value of π is negative eight. We therefore have that π of π₯ is equal to ππ₯ squared minus eight. But we need to determine the value of π. To do this, we can use the coordinates of any other point on the curve.

So letβs consider the point with coordinates three, one. In other words, when π₯ is equal to three, π of π₯ must be equal to one. Substituting π₯ is equal to three and π of π₯ is equal to one into our equation, we have one is equal to π multiplied by three squared minus eight. That simplifies to one is equal to nine π minus eight. And then we add eight to each side of the equation to give nine is equal to nine π. By dividing both sides of this equation by nine, it follows that π is equal to one.

So we know that π of π₯ is of the form ππ₯ squared plus π. And we found that π is equal to one and π is equal to negative eight. So the equation of the function drawn on the graph is π of π₯ equals π₯ squared minus eight, which is option (b).

Letβs just briefly consider the four other options we were given. Looking at option (e) π of π₯ equals π₯ minus eight, this represents the linear function. So the graph of this function would be a straight line. And so we know straight away that this one is not the correct function. Looking at options (a) and (d), the value of π, the constant term in each function, is positive eight. And so these two functions would each have a π¦-intercept of positive, not negative, eight. Looking at option (c), this does have the correct π¦-intercept, but the coefficient of π₯ squared is negative one when it should be positive one. In fact, this function would be a parabola with a π¦-intercept of negative eight, but as the coefficient of π₯ squared is negative, it would be a parabola that opens downwards.

We can of course check our answer by picking any other point on the graph, for example, the point with coordinates negative four, eight. Substituting π₯ equals negative four into function (b), we have π of negative four equals negative four squared minus eight. Thatβs 16 minus eight, which is equal to eight. And so this confirms that our answer is correct. The equation of the function drawn on the graph is π of π₯ equals π₯ squared minus eight.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions