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What is the general form of a quadratic equation?
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The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants.
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What does the graph of a quadratic function represent?
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The graph of a quadratic function represents a parabola.
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How can you tell if a parabola opens upwards or downwards?
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If the coefficient of x^2 (a) is positive, the parabola opens upwards. If a is negative, it opens downwards.
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What is the vertex of a parabola?
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The vertex is the highest or lowest point on the parabola, depending on its orientation.
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How do you find the x-coordinate of the vertex of a parabola given by y = ax^2 + bx + c?
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The x-coordinate of the vertex is given by -b/(2a).
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What are the roots of a quadratic equation?
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Roots of a quadratic equation are the x-values where the graph intersects the x-axis (y=0).
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How are the roots of a quadratic equation represented on the graph?
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The roots are represented as the points where the parabola crosses the x-axis.
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What does it mean if a parabola does not intersect the x-axis?
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If a parabola does not intersect the x-axis, the quadratic equation has no real roots.
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What is the axis of symmetry in a parabola?
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The axis of symmetry is a vertical line that passes through the vertex of the parabola, given by x = -b/(2a).
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How does the discriminant affect the number of roots?
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The discriminant (b^2 - 4ac) determines the number and type of roots: >0 means 2 real and distinct roots; =0 means 1 real repeated root; <0 means no real roots.
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How can you graph a quadratic equation by using its vertex and axis of symmetry?
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Plot the vertex using x = -b/(2a) and find additional points using symmetry about the axis.
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Can a quadratic equation have two different y-values for a single x-value? Why or why not?
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No, a quadratic equation is a function, which means each x-value has exactly one corresponding y-value.
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What role does the constant c play in the graph of a quadratic equation?
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The constant c represents the y-intercept, the point where the parabola crosses the y-axis.
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How is completing the square helpful for graphing quadratic equations?
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Completing the square transforms a quadratic equation into vertex form, making it easier to identify the vertex and graph the equation.
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What information do you need to graph a standard quadratic equation?
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You need the vertex, axis of symmetry, direction of opening (based on a), and y-intercept.
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How can transformations affect the graph of a quadratic function?
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Transformations include shifts (horizontal and vertical), reflections (across the x-axis), and scaling (stretching or compressing).
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What is the significance of the y-intercept of a quadratic graph?
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The y-intercept indicates the value of the function when x = 0, marking the point where the parabola crosses the y-axis.
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How do you determine the width of a parabola?
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The width of a parabola is affected by the absolute value of a; smaller values result in wider parabolas, and larger values lead to narrower parabolas.
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What happens to the graph of a quadratic function if you add a constant k to the equation y = ax^2 + bx + c?
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Adding a constant k results in a vertical shift of the entire parabola. If k is positive, the shift is upward; if negative, the shift is downward.
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Why is zero worth noting on the axis of symmetry?
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The axis of symmetry passes through the vertex and divides the parabola into two mirror-image halves, showing the balance of the graph.
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