graph the function and state the vertex axis of symmetry, the intercepts, if any: f(x)=x^2-8x+15
Please show work so i can understand this. thank you
F(x) = Y = x^2 - 8x + 15.
h = Xv = -B/2A = 8 / 2 = 4.
K = Yv = 4^2 - 8*4 + 15 = -1.
V(h,k) = V(4,-1).
Axis = h = 4.
The intercepts are the value of X when
Y = 0:
Y = x^2 - 8x + 15 = 0
C = 15 = (-1)*(-15) = (-3)*(-5).
We select -3 and -5 for factoring because the sum = B(-8).
(x-5)(x-3) = 0
x-5 = 0
X = 5.
x-3 = 0
X = 3.
Intercepts(Solution set): X = 5, and 3.
To graph the function, select values of X below and above h; and calculate the
corresponding value of Y.
(x,y). Y = x^2 - 8x + 15.
(1,8).
(2,3).
(3,0).
V(4,-1).
(5,0).
(6,3).
(7,8).
To graph the function f(x) = x^2 - 8x + 15, we can follow a few steps:
Step 1: Find the vertex of the parabola. The vertex formula is given by x = -b/2a, where a and b are the coefficients of x^2 and x, respectively. In this case, a = 1 and b = -8. Plugging these values into the formula, we get x = -(-8)/(2*1) = 8/2 = 4. This gives us the x-coordinate of the vertex.
Step 2: To find the y-coordinate of the vertex, substitute the x-coordinate we found (x = 4) into the equation f(x) = x^2 - 8x + 15. Evaluating it, we have f(4) = (4)^2 - 8(4) + 15 = 16 - 32 + 15 = -1. Therefore, the vertex is (4, -1).
Step 3: Determine the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex. In this case, the equation of the axis of symmetry is x = 4.
Step 4: Find the y-intercept. To find the y-intercept, substitute x = 0 into the function f(x) = x^2 - 8x + 15. Evaluating it, we have f(0) = (0)^2 - 8(0) + 15 = 0 - 0 + 15 = 15. So, the y-intercept is (0, 15).
Step 5: Find the x-intercepts. To find the x-intercepts, we need to solve the equation f(x) = 0. In this case, we have x^2 - 8x + 15 = 0. Factoring or using the quadratic formula, we find that (x - 3)(x - 5) = 0. Therefore, the x-intercepts are x = 3 and x = 5. So, the x-intercepts are (3, 0) and (5, 0).
Now that we have all the information, we can graph the function:
- Plot the vertex at (4, -1).
- Draw the axis of symmetry x = 4, which is a vertical line passing through the vertex.
- Plot the y-intercept at (0, 15).
- Plot the x-intercepts at (3, 0) and (5, 0).
- Finally, draw a smooth U-shaped curve passing through the vertex and the intercepts.
The graph of the function f(x) = x^2 - 8x + 15 should resemble an upward-opening parabola with the vertex (4, -1), the axis of symmetry x = 4, the y-intercept (0, 15), and the x-intercepts (3, 0) and (5, 0).