Quadratic Inequalities & Real World Applications

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What is a quadratic inequality?

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A quadratic inequality is an inequality that involves a quadratic expression, typically in the form ax^2 + bx + c < 0, ax^2 + bx + c > 0, ax^2 + bx + c ≤ 0, or ax^2 + bx + c ≥ 0.

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How do you determine the direction of a parabola given a quadratic equation?

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The direction of a parabola is determined by the coefficient of the x^2 term. If the coefficient is positive, the parabola opens upwards; if it is negative, it opens downwards.

Tags

CCSS.HSF-IF.C.7A

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What is the vertex of a parabola, and why is it important?

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The vertex of a parabola is the highest or lowest point on the graph, depending on whether it opens up or down. It is important because it represents the maximum or minimum value of the quadratic function.

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How can you find the maximum height of a projectile modeled by a quadratic function?

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To find the maximum height, you can use the vertex formula x = -b/(2a) to find the time at which the maximum height occurs, then substitute that time back into the function to find the height.

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

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The discriminant (b^2 - 4ac) determines the nature of the roots of the quadratic equation. If it is positive, there are two distinct real roots; if it is zero, there is one real root; if it is negative, there are no real roots.

Tags

CCSS.HSA-REI.B.4B

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How do you solve a quadratic inequality graphically?

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To solve a quadratic inequality graphically, you first graph the corresponding quadratic equation, identify the x-intercepts, and then determine the intervals where the quadratic expression is above or below the x-axis.

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What is the standard form of a quadratic function?

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The standard form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0.

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What is the vertex form of a quadratic function?

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The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

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How do you find the time when a projectile reaches a certain height?

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To find the time when a projectile reaches a certain height, set the height equation equal to the desired height and solve for the variable representing time.

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What is the formula for the height of an object in free fall?

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The height of an object in free fall can be modeled by the equation h(t) = -16t^2 + vt + h0, where v is the initial velocity and h0 is the initial height.