1.6 Quadratics - Finding the Solutions From a Graph

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What is a quadratic function?

Back

A quadratic function is a polynomial function of degree 2, typically written in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0.

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What does it mean for a quadratic function to have 2 solutions?

Back

A quadratic function has 2 solutions when its graph intersects the x-axis at two distinct points.

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What does it mean for a quadratic function to have 1 solution?

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A quadratic function has 1 solution when its graph touches the x-axis at exactly one point, known as a double root.

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What does it mean for a quadratic function to have no solution?

Back

A quadratic function has no solution when its graph does not intersect the x-axis at all, indicating that the quadratic expression is always positive or always negative.

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How can you find the solutions of a quadratic function from its graph?

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You can find the solutions by identifying the x-coordinates of the points where the graph intersects the x-axis.

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What is the vertex of a quadratic function?

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The vertex is the highest or lowest point on the graph of a quadratic function, depending on the direction of the parabola.

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What is the axis of symmetry in a quadratic function?

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The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves, passing through the vertex.

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What is the significance of the discriminant in a quadratic equation?

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The discriminant (b² - 4ac) determines the number of solutions: if it's positive, there are 2 solutions; if zero, 1 solution; if negative, no solution.

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What does it mean if a quadratic function has an empty set of solutions?

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An empty set of solutions means that there are no x-values for which the quadratic function equals zero, indicating no x-intercepts.

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How do you determine if a quadratic function opens upwards or downwards?

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A quadratic function opens upwards if the coefficient 'a' is positive and downwards if 'a' is negative.