identify the ff properties in the given quadratic equations:
a.Vertex b. axis of symmetry c. Domain
d. Range e direction of graph
1.f(x)=x2-5x+14
complete the square:
f(x)=x^2-5x+25/4 + 14-25/4
= (x-5/2)^2 + 7.75
so the vertex is at x=5/2, y=7.75
axis of symmetry x=5/2
domain all real x you do the rest. On the direction of the graph, what happens as x gets very large?
To identify the properties of the given quadratic equation, f(x) = x^2 - 5x + 14, we need to analyze the equation and use formulas and principles related to quadratic functions.
a. Vertex: The vertex of a quadratic equation is given by the formula (-b/2a, f(-b/2a)). In our equation, a = 1, b = -5, and c = 14. Thus, the vertex is (-(-5)/(2*1), f(-(-5)/(2*1))). Simplifying the equation, we have (5/2, f(5/2)).
To find the value of f(5/2), we substitute it into the original quadratic equation:
f(5/2) = (5/2)^2 - 5(5/2) + 14
= 25/4 - 25/2 + 14
= 25/4 - 50/4 + 56/4
= 31/4
Therefore, the vertex is found at (5/2, 31/4).
b. Axis of symmetry: The axis of symmetry of a quadratic equation is a vertical line that passes through the vertex. The equation of the axis of symmetry can be determined by using the x-value from the vertex. In this case, the axis of symmetry is x = 5/2.
c. Domain: The domain is the set of all possible x-values for which the equation is defined. In this case, since we have a quadratic equation, the domain is all real numbers since we can input any value for x.
d. Range: The range is the set of all possible y-values that the equation can produce. In this case, since the coefficient of the quadratic term (x^2) is positive, the parabola opens upwards, which means it has a minimum point (the vertex). Hence, the range starts from the y-coordinate of the vertex and goes to positive infinity. Therefore, the range is [31/4, ∞).
e. Direction of graph: By looking at the coefficient of the quadratic term (x^2), we can determine the direction of the graph. In this case, the coefficient is positive, so the parabola opens upwards.
In summary:
a. Vertex: (5/2, 31/4)
b. Axis of symmetry: x = 5/2
c. Domain: All real numbers
d. Range: [31/4, ∞)
e. Direction of graph: Upwards opening parabola.