Quadratic functions
54 flashcards covering Quadratic functions for the SAT Math section.
Quadratic functions are mathematical expressions that describe relationships forming a U-shaped curve, known as a parabola. Essentially, they take the form y = ax² + bx + c, where a, b, and c are numbers, and the squared term makes the graph curve rather than form a straight line. These functions are useful for modeling real-world scenarios, like the path of a thrown ball or the growth of profits over time, helping us predict outcomes based on patterns.
On the SAT Math section, quadratic functions appear in algebra problems that test your ability to solve equations, graph parabolas, find vertices, or interpret word problems involving maximums and minimums. Common traps include mistaking linear equations for quadratics or errors in factoring, so watch for questions that require the quadratic formula or completing the square. Focus on understanding how to manipulate these functions and apply them to practical contexts, as the test often combines them with other topics like systems of equations.
Practice factoring quadratics quickly to save time on multiple-choice questions.
Terms (54)
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Quadratic function
A quadratic function is a polynomial function of degree two, generally written as f(x) = ax² + bx + c, where a, b, and c are constants and a is not zero.
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Standard form of a quadratic
The standard form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are real numbers and a ≠ 0, allowing for easy identification of coefficients.
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Vertex form of a quadratic
The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola, making it useful for graphing and finding the maximum or minimum.
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Factored form of a quadratic
The factored form of a quadratic function is f(x) = a(x - r)(x - s), where r and s are the roots, which helps in determining the x-intercepts of the graph.
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Parabola
A parabola is the U-shaped graph of a quadratic function, which opens upward if a > 0 and downward if a < 0, and it is symmetric about its axis.
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Axis of symmetry
The axis of symmetry of a quadratic function is the vertical line that passes through the vertex, given by x = -b/(2a) for f(x) = ax² + bx + c.
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Vertex of a parabola
The vertex of a parabola is the highest or lowest point on the graph of a quadratic function, representing the maximum if a < 0 or minimum if a > 0.
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Roots of a quadratic
The roots of a quadratic equation are the values of x that make f(x) = 0, also known as zeros, and they correspond to the x-intercepts of the parabola.
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Discriminant
The discriminant of a quadratic equation ax² + bx + c = 0 is b² - 4ac, which determines the nature of the roots: positive for two real roots, zero for one real root, and negative for no real roots.
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Quadratic formula
The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a) for an equation ax² + bx + c = 0, providing the roots regardless of whether the quadratic factors easily.
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Factoring a quadratic
Factoring a quadratic involves rewriting ax² + bx + c as a product of binomials, such as (x + p)(x + q), to solve for roots or simplify expressions.
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Completing the square
Completing the square is a method to rewrite a quadratic in vertex form by adding and subtracting a constant to ax² + bx + c, making it easier to find the vertex or solve equations.
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Sum of roots
For a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a, which can be used to verify solutions or analyze the equation without finding the roots explicitly.
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Product of roots
For a quadratic equation ax² + bx + c = 0, the product of the roots is c/a, helpful for checking factored forms or understanding the equation's behavior.
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Graph of a quadratic function
The graph of a quadratic function is a parabola that shows how the function's values change with x, including key features like the vertex, axis of symmetry, and intercepts.
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Maximum value of a quadratic
The maximum value of a quadratic function occurs at the vertex when a < 0, and it is the y-coordinate of that point, often found in optimization problems.
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Minimum value of a quadratic
The minimum value of a quadratic function occurs at the vertex when a > 0, representing the lowest point on the parabola in real-world applications.
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Domain of a quadratic function
The domain of a quadratic function is all real numbers, meaning x can be any value, as there are no restrictions like division by zero or square roots of negatives in the basic form.
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Range of a quadratic function
The range of a quadratic function is from the vertex's y-value to infinity if it opens upward, or from negative infinity to the vertex's y-value if it opens downward.
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Solving quadratic equations
Solving quadratic equations involves methods like factoring, completing the square, or using the quadratic formula to find the values of x that satisfy ax² + bx + c = 0.
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Quadratic inequality
A quadratic inequality compares a quadratic expression to zero, such as ax² + bx + c > 0, and is solved by finding the roots and testing intervals on the number line.
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Vertex formula
The vertex formula gives the x-coordinate of the vertex as x = -b/(2a) for f(x) = ax² + bx + c, allowing quick location of the parabola's turning point.
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Y-intercept of a quadratic
The y-intercept of a quadratic function is the value of f(0), which is c in the standard form ax² + bx + c, representing where the graph crosses the y-axis.
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X-intercepts of a quadratic
The x-intercepts of a quadratic are the points where the graph crosses the x-axis, corresponding to the roots of the equation ax² + bx + c = 0.
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Symmetry in quadratics
Symmetry in quadratics means the graph is mirror-like across the axis of symmetry, so points equidistant from this line have the same y-values.
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Difference of squares
The difference of squares is a factoring technique for expressions like a² - b², which factors to (a - b)(a + b), useful for simplifying quadratics.
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Perfect square trinomial
A perfect square trinomial is an expression like a² + 2ab + b², which factors to (a + b)², and recognizing it aids in completing the square or factoring.
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Using substitution in quadratics
Substitution in quadratics involves replacing a complex expression with a single variable to simplify solving, such as letting u = x² for nested quadratics.
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Word problems: projectile motion
In projectile motion problems, a quadratic equation models the height of an object over time, with the vertex representing the maximum height.
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Word problems: area and dimensions
Area and dimensions problems often use quadratics to represent scenarios like maximizing area given a perimeter, solved by finding the vertex.
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Systems: quadratic and linear
Solving a system with a quadratic and a linear equation involves finding intersection points, which can be done by substitution or graphing.
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Positive discriminant
A positive discriminant indicates two distinct real roots for the quadratic equation, meaning the parabola crosses the x-axis at two points.
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Negative discriminant
A negative discriminant means no real roots, so the quadratic equation has two complex solutions and the parabola does not cross the x-axis.
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Zero discriminant
A zero discriminant indicates exactly one real root, meaning the quadratic touches the x-axis at a single point, which is the vertex.
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Strategy: factoring vs. quadratic formula
When solving quadratics, try factoring first if the equation is simple, but use the quadratic formula for messy coefficients to ensure accuracy.
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Common trap: extraneous solutions
Extraneous solutions can occur when solving quadratics by methods like square roots, so always verify solutions by plugging them back into the original equation.
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Common trap: sign errors in factoring
Sign errors in factoring quadratics often happen with the middle term, so double-check that the binomial factors multiply correctly to the original expression.
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Example: solving by factoring
For the equation x² + 5x + 6 = 0, factor it as (x + 2)(x + 3) = 0, so the solutions are x = -2 and x = -3.
Solve x² - 4x + 4 = 0 by factoring: (x - 2)² = 0, so x = 2.
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Example: solving by quadratic formula
For 2x² + 3x - 2 = 0, use the quadratic formula: x = [-3 ± √(9 + 16)] / 4 = [-3 ± 5]/4, yielding x = 0.5 and x = -2.
For x² - 2x - 3 = 0, x = [2 ± √(4 + 12)] / 2 = [2 ± √16]/2 = [2 ± 4]/2, so x = 3 or x = -1.
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Example: completing the square
To solve x² + 6x - 7 = 0, add 9 to both sides: x² + 6x + 9 = 16, so (x + 3)² = 16, giving x + 3 = ±4, thus x = 1 or x = -7.
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Example: finding vertex from standard form
For f(x) = -x² + 4x - 1, the vertex is at x = -b/(2a) = -4/(2(-1)) = 2, and f(2) = -1, so vertex is (2, -1).
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Example: graphing a quadratic
For f(x) = x² - 4x + 3, find vertex at (2, -1), roots at x=1 and x=3, and y-intercept at 3, then sketch the upward-opening parabola.
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Transformation of quadratics
Transformations of quadratics include shifts, stretches, and reflections, such as f(x) = a(x - h)² + k, where h and k shift the graph horizontally and vertically.
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Reflection of quadratics
A reflection of a quadratic occurs when a is negative, flipping the parabola upside down compared to the basic y = x².
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Stretching/compressing quadratics
In f(x) = a x² + b x + c, the absolute value of a determines stretching if |a| > 1 or compressing if 0 < |a| < 1 along the y-axis.
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Quadratic in vertex form example
For f(x) = 2(x - 1)² + 3, the vertex is at (1, 3), it opens upward, and the roots are found by solving 2(x - 1)² + 3 = 0.
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Interpreting quadratic coefficients
In f(x) = ax² + bx + c, a affects the width and direction, b influences the axis of symmetry, and c is the y-intercept, all key for analysis.
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Quadratic inequality example
For x² - 4 > 0, solve x² - 4 = 0 to get x = ±2, then test intervals: x < -2 or x > 2 satisfies the inequality.
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Using discriminant in problems
The discriminant helps determine if a quadratic equation has real solutions, which is crucial in contexts like physics for feasible outcomes.
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Factoring by grouping
Factoring by grouping applies to quadratics like x³ + 2x² + x + 2, but for pure quadratics, it's used for expressions that can be grouped into binomials.
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Rational root theorem for quadratics
Though more advanced, for quadratics with integer coefficients, possible rational roots are factors of c over factors of a, aiding in testing solutions.
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Quadratic sequence
A quadratic sequence has second differences constant, like in patterns where terms follow n², useful for identifying quadratic relationships in data.
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Optimization with quadratics
Quadratics are used for optimization, such as maximizing revenue or area, by finding the vertex of the quadratic model.
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Inverse quadratic relations
While quadratics aren't one-to-one, restricting the domain allows finding inverses, though this is less common on the SAT.