Jeffrey is asked to graph f(x)=x^2 - 6x + 5. Below is his graph. Is his graph correct or incorrect?
Prove your answer by discussing the key characteristics of a quadratic function. Include the shape of the graph based on its leading coefficient, calculation of its vertex, maximum or minimum value as well as its x-intercepts. Show your work to receive full credit.

His graph is incorrect, because, Jeffrey drew it with vertex (4 , -4) and x-intercepts (2 , 0) and (5 , 0), but the vertex is (3 , -4) and x-intercepts are (1 , 0) , (5 , 0)

Step-by-step explanation:

The quadratic function y = ax² + bx + c is represented graphically by a parabola

∵ The quadratic is f(x) = x² - 6x + 5

∴ a = 1 , b = -6 and c = 5

∵ The leading coefficient a is positive

∴ The graph of f(x) is a parabola oped upward

∴ The parabola has a minimum vertex

The vertx point of the parabola is (h , k)

∵ h =

∴ h =  

∴ h = 3

- To find k substitute x by 3 in f(x)

∵ f(3) = (3)² - 6(3) + 5 = 9 - 18 + 5

∴ f(3) = -4

∵ k = f(3)

∴ k = -4

That means the coordinates of the vertex of the parabola are (3 , -4)

The x-intercepts means values of x at f(x) = 0

∵ x² - 6x + 5 = 0

- Factorize it into two binomial factors

∵ x² = (x)(x)

∵ 5 = (-1)(-5)

∵ (x)(-1) + (x)(-5) = -x + -5x = -6x

- That means the factors are (x - 1) and (x - 5)

∴ The factors of x² - 6x + 5 are (x - 1) and (x - 5)

∴ (x - 1)(x - 5) = 0

Equate each factor by 0 to find the value of x

∵ x - 1 = 0

- add 1 to both sides

∴ x = 1

∵ x - 5 = 0

- add 5 to both sides

∴ x = 5

∴ The x-intercepts are (1 , 0) and (5 , 0)

That means the parabola intersects the x-axis ate points (1 , 0) and (5 , 0)

Jeffrey drew the graph with vertex (4 , -4) and x-intercepts (2 , 0) and (5 , 0), then his graph is incorrect

there is no question content here, but you for the free points! : d

- "runner b is moving at a slower rate."

- "the graphs shown are graphs of distance(km) against time(min).

the slopes of this graphs are are given by:

slope = (change in y)/(change in x)

= (change in distance)/(change in time)

this is the formula for finding the speed. so, the slope of these graphs represent the speed of the runner.

taking each division in the graph to represent 1 on both axis:

runner a ⇒ speed = (2-0)/(1-0)

= 2/1

= 2 km/min

runner b ⇒ speed = (3-0)/(3-0)

= 3/3

= 1 km/min

this shows that   runner b is moving at a slower rate."