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# Quadratics

 Background The graph of a quadratic expression has a U-shaped graph called a parabola. We can write a quadratic expression in the form ax2 + c, where a is called the leading coefficient. The way the parabola opens depends on the sign of the leading coefficient. In fact, if a is positive, the parabola opens up; if a is negative, the parabola opens down. If the absolute value of a is greater than 1, then the parabola is narrower than y = x2. If the absolute value of a is less than 1, then the parabola is wider than y = x2.  A quadratic function can be written in different forms. Each form gives information about the graph. Standard Form (no linear term) Warm-Up y = -3x2 + 4 matches . â€œThis quadratic function is written in standard form. Because the leading coefficient is negative, the graph opens down. The 3 causes the graph to be narrower than y = x2. The vertex is at (0, 4), because the graph has been shifted upward 4 units. You can test the ordered pair (1, 1). When x = 1, y = -3(1)2 + 4, which is 1.â€ y = ax2 + c Graphical Information c > 0: vertical shift upward of c units c < 0: vertical shift downward of c units Warm-Up y = (x + 3)2 + 1 matches . â€œThis quadratic function can be written in vertex form as The vertex is at The graph opens up because a is positive.â€ Vertex Form y = a(x - h)2 + k Graphical Information vertex at (h, k) Warm-Up matches . â€œThis quadratic function is written in factored form. The graph opens down because a is less than 0. Because the graph is wider than y = x2. The intercepts are at (-5, 0) and (2, 0)â€ Factored Form y = a(x - p)(x - q) Graphical Information x-intercepts at (p, 0) and (q, 0)
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