One of the most powerful skills you can develop in algebra is the ability to look at a graph and immediately determine how many solutions an equation has and what kind of solutions they are. You will learn to read a parabola like a map, using it to answer questions without doing any calculations at all.
1. The Connection Between Equations and Graphs
Every quadratic equation can be connected to a graph. When you have the equation $y = ax^2 + bx + c$ and you graph it, you get a curved shape called a parabola. The solutions to the equation $ax^2 + bx + c = 0$ correspond to the points where the parabola crosses the x-axis.
Why does this work?
Solving $ax^2 + bx + c = 0$ is the same as asking:
"For what values of $x$ does $y$ equal zero?"
And $y$ equals zero exactly on the x-axis.
Real-World Analogy: A Ball in the Air
Imagine you throw a ball upward from a cliff. The height of the ball over time follows a parabolic path. The question "When does the ball hit the ground?" is the same as asking "When does the height equal zero?" On a graph, this corresponds to where the parabola meets the horizontal axis (ground level).
- If the parabola touches the ground once, the ball grazes the ground at one moment.
- If it crosses twice, the ball was on the ground at two different times.
- If it never reaches the ground, the equation has no real solutions.
2. The Three Possible Scenarios
When you look at a parabola and a horizontal line, there are exactly three things that can happen:
Scenario 1
Crosses the line at two points
Two real solutions
Discriminant $> 0$
Scenario 2
Just barely touches the line at one point
One real solution
Discriminant $= 0$
Scenario 3
Never reaches the line
Two non-real solutions
Discriminant $< 0$
| Graph Shows | Solution Type | Discriminant |
|---|---|---|
| Crosses line twice | Two real solutions | Positive ($> 0$) |
| Touches line once | One real solution | Zero ($= 0$) |
| Never reaches line | Two non-real solutions | Negative ($< 0$) |
3. What If the Equation Does Not Equal Zero?
Not all equations are set equal to zero. For instance, you might be asked to solve $x^2 - 6x + 7 = -2$ instead of $x^2 - 6x + 7 = 0$.
The graph of $y = x^2 - 6x + 7$ is the same parabola either way. The only difference is where you look on the graph:
When the equation equals zero, look at where the parabola crosses the x-axis ($y = 0$).
When the equation equals $-2$, look at where the parabola reaches the horizontal line $y = -2$.
The principle is exactly the same: count the intersections to determine the number of real solutions.
Worked Example: $y = x^2 - 6x + 7$
Suppose the vertex (lowest point) of this parabola sits at $y = -2$.
$x^2 - 6x + 7 = 0$
Look where the graph crosses $y = 0$ (the x-axis). The parabola crosses it at two points, so there are two real solutions.
$x^2 - 6x + 7 = -2$
Look where the graph reaches $y = -2$. The vertex just touches that line at one point, so there is one real solution (a repeated root).
$x^2 - 6x + 7 = -5$
Look where the graph reaches $y = -5$. The parabola never dips that low (its lowest point is $y = -2$), so there are zero real solutions and two non-real solutions.
4. Upward vs. Downward Opening Parabolas
Not all parabolas open upward. If the leading coefficient (the number in front of $x^2$) is negative, the parabola opens downward like an upside-down U. For these parabolas, the vertex is the highest point instead of the lowest.
Opens Upward ($a > 0$)
- • Vertex is the lowest point
- • Can't reach $y$-values below the vertex
Opens Downward ($a < 0$)
- • Vertex is the highest point
- • Can't reach $y$-values above the vertex
When a parabola opens downward, the logic is reversed. If you are looking for where the graph reaches a $y$-value that is above the vertex, the parabola will never reach it, giving you no real solutions.
Example
If a downward-opening parabola peaks at $y = 4$ and you need $y = 6$, the parabola cannot reach that high. The equation would have no real solutions.
Remember This Rule
A quadratic equation always has exactly two solutions in total (counting multiplicity). If there are not two real solutions, the missing ones are non-real. The three possibilities:
- 2 real + 0 non-real
- 1 real (repeated) + 0 non-real
- 0 real + 2 non-real
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