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Describe Quadratic Parabola Graphs Using Discriminant and Coefficient a Values
Mathematics
Grade 10 (Junior High School)
Question Content
5. Describe the graph given the following information.\na. The value of the discriminant is -6 and a>0.\nb. The value of the discriminant is 49 and a<0.\nc. The value of the discriminant is 0 and a>0.
Correct Answer
a. The graph is a parabola opening upwards that does not intersect the x-axis.\nb. The graph is a parabola opening downwards that intersects the x-axis at two distinct real points.\nc. The graph is a parabola opening upwards that touches the x-axis at exactly one real point (a tangent point).
Detailed Solution Steps
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Step 1: Recall the properties of quadratic function graphs (parabolas): For a quadratic function in the form $y=ax^2+bx+c$, $a$ determines the direction the parabola opens (positive $a$ opens upwards, negative $a$ opens downwards), and the discriminant $D=b^2-4ac$ determines the number of x-intercepts.
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Step 2: Analyze part a: Given $D=-6<0$, the quadratic has no real roots, so the parabola does not cross the x-axis. Since $a>0$, the parabola opens upwards. Combine these to describe the graph.
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Step 3: Analyze part b: Given $D=49>0$, the quadratic has two distinct real roots, so the parabola intersects the x-axis at two points. Since $a<0$, the parabola opens downwards. Combine these to describe the graph.
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Step 4: Analyze part c: Given $D=0$, the quadratic has exactly one real repeated root, so the parabola touches the x-axis at a single tangent point. Since $a>0$, the parabola opens upwards. Combine these to describe the graph.
Knowledge Points Involved
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Quadratic Function Graph Direction
For a quadratic function $y=ax^2+bx+c$, the coefficient $a$ controls the opening direction of the parabola: if $a>0$, the parabola opens upwards (U-shaped); if $a<0$, the parabola opens downwards (inverted U-shaped). This property determines the overall shape and whether the function has a minimum or maximum value.
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Discriminant of a Quadratic Function
The discriminant is calculated using the formula $D=b^2-4ac$ for a quadratic function $y=ax^2+bx+c$. It is used to determine the number of real x-intercepts (roots) of the quadratic: $D>0$ means two distinct real roots, $D=0$ means one repeated real root, and $D<0$ means no real roots.
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Relationship Between Discriminant and Parabola X-Intercepts
The value of the discriminant directly corresponds to how the parabola interacts with the x-axis: positive discriminant means two crossing points, zero discriminant means a single tangent point, and negative discriminant means no intersection with the x-axis. This is a key connection between algebraic calculations and graphical representations of quadratics.
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