Simple Quadratic Equations
A quadratic equation has an x² term, and its graph is a U-shaped curve called a parabola. In y = ax² (a ≠ 0), the sign of a decides if it opens up or down. The size of a sets the width.
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What Is a Quadratic Equation?
- A quadratic equation has an x² term.
- Its graph is called a parabola.
Graphing the Standard Parabola y = x²
- You find points by squaring x to get y.
- The graph is symmetrical and has a minimum at the origin.
- Points like (-2,4) and (2,4) show the symmetry.
Effect of a in y = ax²
- The sign of a decides if the parabola opens upwards or downwards.
- The size of |a| controls the width of the parabola.
- A larger |a| makes the parabola narrower.
Practice Questions
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What happens when a is positive in a quadratic equation?
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When a is positive, the parabola opens upward.
What can you tell about the parabola if a is negative in a quadratic equation?
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When a is negative, the parabola opens downward. The width of the parabola depends on the absolute value of a. Without knowing the exact value of a, we cannot determine if the parabola is wide or narrow.
If , how does the parabola change compared to when ?
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When , the parabola opens upward because a is positive. Since the absolute value of a is greater than 1, the parabola becomes narrower compared to when .
If , how does the parabola change compared to when ?
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When , the parabola opens upward because a is positive. Since the absolute value of a is less than 1, the parabola is wider compared to when .
If , how does the parabola change compared to when ?
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When , the parabola opens downward because a is negative. Since the absolute value of a is greater than 1, the parabola is narrower compared to when .
If , how does the parabola change compared to when ?
Correct! 🎉 +30 pointsNot quite right
When , the parabola opens downward because a is negative. Since the absolute value of a is less than 1, the parabola is wider compared to when .
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Interactive Activity
Visualise the quadratic function y = ax², where a cannot be 0
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Students Also Ask
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A parabola is symmetric about the y-axis because squaring x and squaring its negative give the same result. For example, (-2)² and 2² both equal 4. Each x-value and its negative share a y-value, so the left and right halves mirror each other.
When the coefficient a is negative, the parabola opens downward instead of upward. The sign of a sets the opening direction, so a negative value flips the curve over. For example, when a = -1, the point at x = 1 drops to y = -1, below the x-axis.
The vertex of y = x² sits at the origin, the point (0, 0). This happens because when x = 0, y also equals 0. The vertex is the turning point of the parabola, and for the standard parabola it rests exactly where the axes cross.
The size of a sets how wide or narrow the parabola is. The further a sits from zero, the narrower the curve becomes. The closer a sits to zero, the wider it spreads, flattening towards the x-axis. So a = 4 gives a narrow curve and a = 0.1 a wide one.
A quadratic graph curves because of its x² term. Squaring the x-values makes them grow quickly as x moves away from zero, so the points climb steeply on each side. Joining them produces a smooth curve, the parabola, rather than a straight line.
Substitute x = 1 into y = ax² and you get y = a. This single point shows the shape at a glance. A higher value of a means a narrower curve, a value near zero means a wider one, and a negative value means the parabola opens downward.