Being able to look at a quadratic equation and identify which graph it corresponds to is like being able to look at a recipe and know what the finished dish will look like. In this section, you will learn the systematic process for connecting equations to their visual representations.
1. The Two-Step Matching Process
When you are given several equations and several graphs, the process for matching them always follows two simple steps.
Step 1: Match the Left Side to the Correct Graph
Each graph is labeled with its equation, such as $y = x^2 - 6x + 7$ or $y = 3x^2 + 2x + 1$. When you look at an equation like $x^2 - 6x + 7 = 0$, the expression on the left side ($x^2 - 6x + 7$) tells you which graph to use. Simply find the graph whose label matches the left side of your equation.
Step 2: Use the Right Side to Know Where to Look
The number on the right side of the equation tells you which horizontal line to examine on the graph:
- If the equation equals $0$, look at the x-axis ($y = 0$)
- If the equation equals $-2$, imagine a horizontal line at $y = -2$
- If the equation equals $6$, look at the horizontal line $y = 6$
Count how many times the parabola crosses or touches that line.
Real-World Analogy: A Building Directory
Think of a tall building with an elevator. The graphs are like the elevator shafts — each one is a different shaft in a different building. The left side of the equation tells you which building to go to. The right side tells you which floor to check. Once you are in the right building and on the right floor, you simply look and count how many rooms (intersections) are on that floor.
2. A Complete Worked Example
Say you are given three graphs labeled:
$y = x^2 - 6x + 7$
Opens upward
$y = 3x^2 + 2x + 1$
Opens upward, narrow, sits above x-axis
$y = -x^2 - 3x + 2$
Opens downward
| Equation | Which Graph? | Look at $y = ?$ | Result |
|---|---|---|---|
| $x^2 - 6x + 7 = 0$ | $y = x^2 - 6x + 7$ | $y = 0$ | 2 real solutions |
| $3x^2 + 2x + 1 = 0$ | $y = 3x^2 + 2x + 1$ | $y = 0$ | 0 real, 2 non-real |
| $-x^2 - 3x + 2 = 0$ | $y = -x^2 - 3x + 2$ | $y = 0$ | 2 real solutions |
| $x^2 - 6x + 7 = -2$ | $y = x^2 - 6x + 7$ | $y = -2$ | 1 real solution |
| $-x^2 - 3x + 2 = 6$ | $y = -x^2 - 3x + 2$ | $y = 6$ | 0 real, 2 non-real |
| $3x^2 + 2x + 1 = 2$ | $y = 3x^2 + 2x + 1$ | $y = 2$ | 2 real solutions |
3. Key Observations
The Same Graph Can Give Different Answers
Notice that the graph $y = 3x^2 + 2x + 1$ gave us zero real solutions when the equation equaled $0$, but two real solutions when it equaled $2$. This is because the parabola sits entirely above the x-axis but does cross the line $y = 2$.
The graph itself did not change. Only the question we were asking about it changed.
Downward Parabolas Have an Upper Limit
The graph of $y = -x^2 - 3x + 2$ gave two real solutions when the equation equaled $0$ (the parabola crosses the x-axis twice) but zero real solutions when the equation equaled $6$ (the parabola never reaches that high, since its peak is below $y = 6$).
The Big Takeaway
The number and type of solutions are determined not just by the shape of the graph, but by the specific y-value you are investigating.
Key Insight
When a problem says "use the graph" to find the number of solutions, you are being asked to be a visual detective. Your eyes are doing the work that the discriminant formula does algebraically. Both methods give the same answer — the graph just lets you see it.
Related Topics
Explore these related math topics: