The graph of 𝑦 is equal to 𝑥 squared plus 𝑏𝑥 plus 𝑐 has been drawn below. Part a) Use the graph to determine which of the following is the value of 𝑐. Circle your answer: is it three, negative four, negative three, or negative one?
When a quadratic equation is of the form 𝑦 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, much like with straight line graphs, the value of 𝑐 represents the 𝑦-intercept. That’s the point of the curve that crosses the 𝑦-axis. Our curve crosses the 𝑦-axis here. We could spend some time considering what the scale of our 𝑦-axis is. But if we look carefully, we can see that the curve crosses the 𝑦-axis exactly halfway between negative two and negative four. The midpoint of negative two and negative four is negative three. So the 𝑦-intercept of our curve is negative three. And that means the value of 𝑐 in turn is negative three.
Part b) Circle the coordinates of the turning point of the graph. Are they one, negative four; negative one, zero; three, zero; or negative four, one? The turning point of the graph is the point at which the graph literally changes direction; it turns. Here it’s a minimum; it’s the lowest possible point on the graph. But sometimes, the turning point would be a maximum. It would be the maximum point on the graph. Let’s begin by finding the value of the 𝑥-coordinate. If we draw a dotted line up until we hit the 𝑥-axis, we can see the 𝑥 coordinate must have a value of one. Similarly, we can draw a horizontal dotted line to find the value of the 𝑦- coordinate. It’s negative four. This means then that the coordinates of our turning point are one, negative four. Be careful negative four, one is there to trip us up. In fact, we always list the 𝑥 value before the 𝑦 value, and we said 𝑥 was one and 𝑦 was negative four.
Part c) Circle the two solutions so the equation 𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. Are they one and negative three, negative one and three, negative one and negative three, or one and three? Originally, we were given a quadratic expression equal to 𝑦. In part c, we’re given the same quadratic expression, this time equal to zero. To get this, we have made 𝑦 be zero; 𝑦 is equal to zero. What does the line 𝑦 equals zero look like? In fact the line 𝑦 equals zero is the 𝑥-axis. And this is because the 𝑦-coordinate of all points on the 𝑥 axis is zero. This means the solutions to the equation 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero are the 𝑥 values of the points where our graph hits the 𝑥-axis. These are sometimes called the roots of the equation. We have one root directly between two and four, so it’s three. The next root falls exactly halfway between zero and negative two, so it’s negative one. We can therefore see that the roots of our equation are negative one and three. The two solutions to the equation are negative one and three.