Algebra 1 (HS pathway)

Quadratic Functions and Equations

Algebra 1

✓By the end of this lesson students will be able to identify and write quadratic functions in standard form.
✓By the end of this lesson students will be able to factor quadratic expressions and solve quadratic equations by factoring.
✓By the end of this lesson students will be able to solve quadratic equations using the quadratic formula.
✓By the end of this lesson students will be able to graph quadratic functions, identifying the vertex, axis of symmetry, and intercepts.
✓By the end of this lesson students will be able to interpret the discriminant to determine the nature and number of solutions.

Key concepts

Quadratic Function

A function that can be written in the standard form f(x) = ax^2 + bx + c, where a, b, and c are real numbers and a ≠ 0. The highest power of the variable is 2. The graph of a quadratic function is a parabola.

f(x) = ax^2 + bx + c

Parabola

The U-shaped graph of a quadratic function. Key features include the vertex (the turning point, either a minimum or maximum), the axis of symmetry (a vertical line through the vertex that divides the parabola into two mirror images), x-intercepts (where the parabola crosses the x-axis, also called roots or zeros), and the y-intercept (where the parabola crosses the y-axis).

Factoring Quadratic Expressions

The process of rewriting a quadratic expression as a product of two or more linear expressions. This method is used to solve quadratic equations by applying the Zero Product Property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero. Common factoring techniques include finding two numbers that multiply to 'ac' and add to 'b' (for ax^2 + bx + c), difference of squares (a^2 - b^2 = (a-b)(a+b)), and greatest common factor.

Quadratic Formula

A formula used to find the solutions (roots or x-intercepts) of any quadratic equation in the standard form ax^2 + bx + c = 0. It is a universal method for solving quadratic equations, even those that are not easily factorable. The discriminant, b^2 - 4ac, within the formula indicates the number and type of real solutions.

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

Discriminant

The expression b^2 - 4ac from the quadratic formula. It determines the nature and number of real solutions to a quadratic equation ax^2 + bx + c = 0:\n* If b^2 - 4ac > 0, there are two distinct real solutions.\n* If b^2 - 4ac = 0, there is one real solution (a repeated root).\n* If b^2 - 4ac < 0, there are no real solutions (two complex solutions).

D = b^2 - 4ac

Key facts to remember

1The standard form of a quadratic function is f(x) = ax^2 + bx + c, where a ≠ 0.
2The graph of a quadratic function is a parabola. If a > 0, it opens up; if a < 0, it opens down.
3The x-coordinate of the vertex of a parabola is given by the formula x = -b / (2a).
4The axis of symmetry is the vertical line x = -b / (2a).
5The quadratic formula, x = (-b ± sqrt(b^2 - 4ac)) / (2a), solves any quadratic equation in the form ax^2 + bx + c = 0.
6The discriminant, D = b^2 - 4ac, indicates the number and type of real solutions: D > 0 (two real), D = 0 (one real), D < 0 (no real).
7To find x-intercepts (roots/zeros), set f(x) = 0. To find the y-intercept, set x = 0.
8The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero.

Worked examples

Example 1

Solve the equation x^2 - 5x = 14 by factoring.

IRewrite the equation in standard form ax^2 + bx + c = 0: x^2 - 5x - 14 = 0.

IIIdentify two numbers that multiply to c (-14) and add to b (-5). These numbers are -7 and 2.

IIIFactor the quadratic expression: (x - 7)(x + 2) = 0.

IVApply the Zero Product Property by setting each factor equal to zero: x - 7 = 0 or x + 2 = 0.

VSolve each linear equation: x = 7 or x = -2.

Answer

x = 7, x = -2

Always ensure the quadratic equation is set to zero before attempting to factor or use the quadratic formula.

Example 2

Solve the equation 2x^2 + 3x - 5 = 0 using the quadratic formula.

IIdentify the values of a, b, and c from the standard form ax^2 + bx + c = 0. Here, a = 2, b = 3, c = -5.

IISubstitute these values into the quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / (2a).

IIIx = (-(3) ± sqrt((3)^2 - 4(2)(-5))) / (2(2))

IVSimplify the expression under the square root (the discriminant): x = (-3 ± sqrt(9 - (-40))) / 4

Vx = (-3 ± sqrt(9 + 40)) / 4

VIx = (-3 ± sqrt(49)) / 4

VIIx = (-3 ± 7) / 4

VIIICalculate the two possible solutions:\n x1 = (-3 + 7) / 4 = 4 / 4 = 1\n x2 = (-3 - 7) / 4 = -10 / 4 = -5/2

Answer

x = 1, x = -5/2

The quadratic formula can solve any quadratic equation, even those that are not easily factorable. Pay close attention to signs when substituting values.

Example 3

Graph the quadratic function f(x) = x^2 - 4x + 3. Identify the vertex, axis of symmetry, x-intercepts, and y-intercept.

I**Find the vertex**: The x-coordinate of the vertex is given by x = -b / (2a). For f(x) = x^2 - 4x + 3, a = 1 and b = -4.\n x = -(-4) / (2 * 1) = 4 / 2 = 2.\n Substitute x = 2 into the function to find the y-coordinate: f(2) = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1.\n Vertex: (2, -1).

II**Find the axis of symmetry**: This is the vertical line passing through the vertex.\n Axis of symmetry: x = 2.

III**Find the y-intercept**: Set x = 0 in the function.\n f(0) = (0)^2 - 4(0) + 3 = 3.\n Y-intercept: (0, 3).

IV**Find the x-intercepts (roots/zeros)**: Set f(x) = 0 and solve for x.\n x^2 - 4x + 3 = 0.\n Factor the quadratic: (x - 1)(x - 3) = 0.\n Set each factor to zero: x - 1 = 0 or x - 3 = 0.\n x = 1 or x = 3.\n X-intercepts: (1, 0) and (3, 0).

V**Plot the points and sketch the parabola**: Plot the vertex (2, -1), the y-intercept (0, 3), its symmetric point (4, 3) (since x=2 is the axis of symmetry), and the x-intercepts (1, 0) and (3, 0). Connect these points with a smooth U-shaped curve. (The graph itself is a visual representation).

Answer

Vertex: (2, -1)\nAxis of Symmetry: x = 2\nY-intercept: (0, 3)\nX-intercepts: (1, 0) and (3, 0)

The sign of 'a' in f(x) = ax^2 + bx + c determines the direction the parabola opens: if a > 0, it opens up; if a < 0, it opens down.

Common mistakes

✗Forgetting to set the quadratic equation to 0 before factoring or applying the quadratic formula.
✗Making sign errors when substituting values for a, b, or c into the quadratic formula, especially with negative values.
✗Incorrectly calculating the discriminant b^2 - 4ac, particularly when b or c are negative.
✗Confusing the x-coordinate of the vertex (-b/2a) with the entire vertex (-b/2a, f(-b/2a)).
✗Not finding a symmetric point for graphing, which can lead to an inaccurate parabola sketch.

Exam tips

★Always check your solutions by substituting them back into the original equation to verify accuracy.
★Understand when each method is most appropriate: factoring for simple, factorable quadratics; the quadratic formula for all quadratics; and graphing for visualizing solutions and key features.
★Show all your steps clearly, especially when using the quadratic formula, to avoid calculation errors and earn partial credit.
★Use a graphing calculator to check your graph's features (vertex, intercepts) but be prepared to show algebraic work for full credit.

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