# Quadratic function passing through two points

by Wayne

Determine the equation of the quadratic function with the given characteristics of it's graph:

x-y coordinates of the vertex: (5,4)

y-intercept 79

 Oct 23, 2013 Equation of Quadratic Function by: Staff Answer Part I The standard form of a quadratic function (a parabola) is: y = ax² + bx + c (i) the value of the constant “c” The quadratic function (a parabola) passes through the y-intercept (0,79). When x = 0, y = 79 To solve for the coefficient “c”, substitute 0 for x and 79 for y in the standard form of the quadratic equation. y = ax² + bx + c 79 = a*0² + b*0 + c 79 = 0 + 0 + c 79 = c c = 79 -------------------------------------------------

 Oct 23, 2013 Equation of Quadratic Function by: Staff ------------------------------------------------- Part II (ii) the value of the coefficient “a” we know the x and y coordinates at the vertex x-y coordinates of the vertex: (5,4) -------------------------------------------------

 Oct 23, 2013 Equation of Quadratic Function by: Staff ------------------------------------------------- Part III substituting these values in the quadratic function, we have an equation with two unknowns (the coefficients a and b) y = ax² + bx + c x = 5 y = 4 c = 79 4 = a*5² + b*5 + 79 4 = 25a + 5b + 79 However, we also know that the first term in the quadratic formula is the x coordinate of the vertex. Not only that, but it contains the same two coefficients: "a" and "b" x coordinate of the vertex = -b/2a x = -b/2a -------------------------------------------------

 Oct 23, 2013 Equation of Quadratic Function by: Staff ------------------------------------------------- Part IV substituting the known value for x (x = 5 at the vertex) 5 = -b/2a solve for b multiply each side of the equation by (-1) (-1)*5 = (-1)*(-b/2a) -5 = b/2a multiply each side of the equation by 2a (2a)*(-5) = (2a)*(b/2a) -10a = b b = -10a -------------------------------------------------

 Oct 23, 2013 Equation of Quadratic Function by: Staff ------------------------------------------------- Part V substitute -10a for b in the equation with the two unknowns 4 = 25a + 5b + 79 4 = 25a + 5*(-10a) + 79 4 = 25a - 50a + 79 4 = -25a + 79 add 25a to each side of the equation and combine like terms 4 + 25a = -25a + 79 + 25a 4 + 25a = -25a + 25a + 79 4 + 25a = 0 + 79 4 + 25a = 79 -------------------------------------------------

 Oct 23, 2013 Equation of Quadratic Function by: Staff ------------------------------------------------- Part VI subtract 4 from each side of the equation 4 + 25a - 4 = 79 - 4 4 - 4 + 25a= 79 - 4 0 + 25a = 79 - 4 25a = 79 - 4 25a = 75 divide each side of the equation by 25 25a/25 = 75/25 (25/25)a = 75/25 (1)a = 75/25 a = 75/25 a = 3 -------------------------------------------------

 Oct 23, 2013 Equation of Quadratic Function by: Staff ------------------------------------------------- Part VII (iii) the value of the coefficient “b” from section (ii) we determined that b = -10a since we computed the value of the coefficient "a" in section (ii), substitute that value for "a" b = -10a a = 3 b = -10 * 3 b = -30 -------------------------------------------------

 Oct 23, 2013 Equation of Quadratic Function by: Staff ------------------------------------------------- Part VIII (iv) the final equation y = ax² + bx + c a = 3 b = -30 c = 79 y = 3*x² + (-30)*x + 79 y = 3x² - 30x + 79 ----------------------------------------------------------------- (v) check your work x-y coordinates of the vertex: (5,4) if the equation is correct, when x = 5, then y should = 4 substitute 5 for x and 4 for y in the final equation y = 3x² - 30x + 79 y = 3*5² - 30*5 + 79 y = 3*25 - 150 + 79 y = 75 - 150 + 79 y = 4, OK y-intercept = 79 if the equation is correct, when x = 0, then y should = 79 substitute 0 for x and 79 for y in the final equation y = 3x² - 30x + 79 y = 3*0² - 30*0 + 79 y = 0 - 0 + 79 y = 79, OK -------------------------------------------------

 Oct 23, 2013 Equation of Quadratic Function by: Staff ------------------------------------------------- Part IX A graph of the equation of the quadratic function is shown below Thanks for writing. Staff www.solving-math-problems.com