The graph of the given function is shown below
The graph is a quadratic function with a minimum at 2.5, negative 2.25 and x intercepts at 1 and 4. The correct option is C
From the question, we are to graph the given quadratic function
The given quadratic function is
f(x) = x² − 5x + 4
The graph of the given function is shown below
The graph is a quadratic function with a minimum at 2.5, negative 2.25 and x intercepts at 1 and 4. The correct option is C
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C) graph of a quadratic function with a minimum at 2.5, negative 2.25 and x intercepts at 1 and 4
Standard Form of a Quadratic Function: ax² + bx + c = 0, where a ≠ 0
Given function: x² - 5x + 4
⇒ a = 1, b = -5, c = 4
The minimum of a function is the lowest point of its parabola. We can use the following formula to find the vertex or the minimum of the function: [tex]x=\dfrac{-b}{2a}[/tex] . This will give us the x-value of the vertex, and we will need to solve for the y-value.
The vertex or minimum.
Step 1: Find the x-value of the vertex.
[tex]\\\implies x=\dfrac{-(-5)}{2(1)}\\\\\implies x=\dfrac{5}{2}=2.5[/tex]
Step 2: Find the y-value of the vertex by substituting 2.5 as the value of x in the given function.
[tex]\\\implies x^2 - 5x + 4\\\\\implies 2.5^2-5(2.5)+4\\\\\implies 6.25-12.5+4\\\\\implies -2.25[/tex]
Vertex: (2.5, -2.25)
The x-intercepts (roots).
We can use the Quadratic Formula to find the roots of this function.
Quadratic Formula: [tex]x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex] when [tex]ax^2+bx+c=0[/tex]
Equation: x² - 5x + 4 = 0
⇒ a = 1, b = -5, c = 4
Step 1: Substitute the values of a, b, and c into the formula.
[tex]\\\implies x=\dfrac{\bold{-(-5)}\pm\sqrt{\bold{(-5)^2}\bold{-4(1)(4)}}}{\bold{2(1)}}\\\\\implies x=\dfrac{5\pm\sqrt{\bold{25-16}}}{2}\\\\\implies x=\dfrac{5\pm\sqrt{\bold{9}}}{2}\\\\\implies x=\dfrac{5\pm3}{2}[/tex]
Step 2: Separate into two possible cases.
[tex]x_1=\dfrac{5-3}{2}\implies \dfrac{2}{2}\implies \boxed{1}\\\\x_2=\dfrac{5+3}{2}\implies \dfrac{8}{2}\implies \boxed{4}[/tex]
The x-intercepts of the given function are 1 and 4.
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