Consider the function 𝑦 equals 𝑎 𝑥 cubed plus three 𝑥 squared plus two 𝑥 plus 𝑑. The graph of the function in the 𝑥𝑦-plane has an 𝑥-intercept at 𝑥 equals three and a 𝑦-intercept at 𝑦 equals one. What is the value of 𝑎?
We can sketch the 𝑥𝑦-plane here, the 𝑥- and 𝑦-axis. We have an 𝑥-intercept at 𝑥 equals three. This is the place where our line crosses the 𝑥-axis. We could label this coordinate three, zero. This function has a 𝑦-intercept at 𝑦 equals one. Here’s this point. The coordinate pair for this point is zero, one. It’s located zero along the 𝑥-axis and one along the 𝑦-axis. We can use these two coordinates to find the value of 𝑎. Notice that the function we were given has two different variables 𝑎 and 𝑑.
Before we can solve for 𝑎, we’ll need to know what 𝑑 equals. 𝑑 is a constant value. It’s not being multiplied by 𝑥. If we use our 𝑦-intercept coordinate, we can solve for 𝑑. At the 𝑦-intercept, 𝑥 equals zero. And 𝑦 equals one. In every place where there is an 𝑥 in our equation, we plug in zero. And where there is 𝑦, we plug in one. Because all of our 𝑥 terms are multiplied by zero, they’re going to be equal to zero.
And then 𝑑 equals one. If 𝑑 equals one, we can use this information and plug it back into the equation we were given, so that it says 𝑦 equals 𝑎 times 𝑥 cubed plus three times 𝑥 squared plus two 𝑥 plus one. And this time we’ll plug in the other coordinate we have to solve for 𝑎. When 𝑥 equals three, 𝑦 equals zero. Three cubed equals 27. And we bring down our 𝑎, three squared equals nine. And nine times three equals 27. Two times three equals six. And bring down that plus one. 27 plus six plus one equals 34. So zero equals 27𝑎 plus 34.
Remember, our goal is to find the value of 𝑎. We need to get 𝑎 by itself. We subtract 34 from both sides. And then we get negative 34 equals 27𝑎. We’ll give ourselves a little bit more room. Negative 34 equals 27𝑎. And so to find the value of 𝑎, we divide both sides by 27. 27𝑎 divided by 27 equals 𝑎. And on the left side, we have negative 34 over 27. 𝑎 equals negative thirty-four twenty-sevenths. Since 34 and 27 don’t have any common factors, this fraction is reduced to its simplest form already. And that’s our final answer. 𝑎 equals negative 34 over 27.