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Quadratic Functions – Intro LINEAR (review) equations look like slope y-intercept form y = mx + b e.g. y = 2x + 7 standard form Ax + By + C = 0 e.g. 2x – y + 7 = 0 other forms: there are other ways to write linear equations… graphs look like y = 2x + 1 (1, 3) (0, 1) note: graphs are always straight lines rise=2 run=1 table of values look like y = 2x + 1 x y first differences 0 1 3–1=2 1 3 5–3=2 2 5 7–5=2 3 7 9–7=2 4 9 note: first differences are constant (don’t change) (i.e. slopes are constant) Quadratic Functions – Intro note: this is the same quadratic, just written 2 different ways QUADRATIC: equations look like standard form y = ax2 + bx + c y = x2 – 7x + 10 factored form y = a(x – s)(x – t) y = (x – 2)(x – 5) note: there are other ways to write quadratic equations… graphs look like note: graphs are always parabolas (“U” shaped) 8 12 4 fx = 10 0.75x-42 -3 vertex (4, 2) x-intercept (2, 0) (one root is 2) y-intercept (0, 9) axis of symmetry x=4 6 2 so, maximum value = 2 so, max value = 2 -5 5 10 6 15 4 x-intercept (6, 0) -2 y-intercept (0, –6) axis of symmetry x=4 8 (one root is 6) x-intercept (6, 0) (one root is 6) 2 -4 -15 -10 -5 5 10 -6 x-intercept (2, 0) -2 (one root is 2) -8 vertex (4, –3) -4 so, minimun value = –3 so, max value = 2 table of values look like note: first differences are changing note: second differences are constant e.g. x –3 –2 –1 0 1 2 3 y = x2 + 2x + 4 y 7 4 3 4 7 12 19 first difference 4 – 7 = –2 3 – 4 = –1 4–3=1 7–4=3 12 – 7 = 5 19 – 12 = 7 second difference –1 – (–2) = 2 1 – (–1) = 2 3 –1 = 2 5–3=2 7–5=2 15 Quadratic Functions – Intro quadratic equations can be in standard form: y = ax2 + bx + c where a and b and c are all numbers but a 0 (can’t be zero), (but b or c can = 0) so, the obvious possibilities are: y = ax2 + bx + c y = ax2 + c y = ax2 can be in the factored form: y = a(x – s)(x – t) y = a(x – s)(x – t) y = a(x – s)(x – t) is ok for a quadratic function is ok for a quadratic function is ok for a quadratic function e.g. y = x2 – 7x + 10 e.g. y = x2 + 10 e.g. y = x2 y = a(x – s)(x – t) where a and s and t are all numbers but a 0 (can’t be zero), (but s or t can = 0) is ok for a quadratic function is ok for a quadratic function is ok for a quadratic function e.g. y = (x – 2)(x – 5) e.g. y = (x – 0)(x – 5) e.g. y = (x – 2)(x – 0) note: the equation can be in other forms as well… note: you can change from standard form to factored form by factoring y = x2 – 7x + 10 y = (x – 2)(x – 5) what are 2 numbers that multiply to +10 and add up to –7 –2 and –5 note: you can change from factored form to standard form by expanding y = (x – 2)(x – 5) use “FOIL” y = x2 – 5x – 2x + 10 y = x2 – 7x + 10 parabola – “U” shaped curve “zeros” vertex (x, y) axis of symmetry y-intercept maximum / minimum value x-intercepts / “roots” / graph of a quadratic must be in the form of a: parabola symmetrical “U” shape vertex – the point on the graph (parabola) with the greatest y-coordinate value (if the graph opens up), OR the point on the graph (parabola) with the least y-coordinate value (if the graph opens down) maximum value – the value of the y-coordinate of the vertex (if the graph opens down) minimum value – the value of the y-coordinate of the vertex (if the graph opens up) axis of symmetry – a (vertical) line through the vertex that divides the parabola into 2 (mirror-image) halves the equation of the axis of symmetry will always be x = some number (the value of the number is the value of the x-coordinate of the vertex) x-intercepts (also known as the “roots” or the “zeros” of the quadratic) – where the parabola crosses the x-axis note: if the parabola opens upwards, and the vertex is above the x-axis, there will be no x-intercepts note: you can determine the x-intercepts (“zeroes”) (“roots”) of a quadratic: 1) by graphing it (by hand or using a graphing calculator) or 2) by factoring it (see previous page) note: the average of the values of the x-intercepts is the value of the x-coordinate of the vertex (x, y) y-intercept – where the parabola crosses the y-axis