Question

f(x) = x² + 10x + 26Opens: Up/DownAxis of Symmetry:Vertex:y-intercept:Number of x-intercept(s):Domain:Range:

Answer 1

Step-by-step explanation:

Given:

f(x) = x² + 10x + 26

To find:

if the equation of up/down, axis of symmetry, vertex, y-intercept, number of x-intercepts

domain, range

i) The sign of the leading coefficient tells us if it opens up or down. The sign is positive. Hence, the parabola opens up

ii) Axis of symmetry is the value of x which splits the parabola into equaloalves (mirror images). The value of x will be the x coordinateof the vertex. Using the graph, the vertex is at (-5, 1)

The x coordinate = 5

At x = 5, the parabola splits into equal halves.

Hence, the axis of symmetry is x = 5

iii) The vertex = (h, k)

`\begin{gathered} h\text{ = }(-b)/(2a) \\ k\text{ = f\lparen h\rparen = f\lparen}(-b)/(2a)) \\ from\text{ the quadratic function:} \\ a\text{ = 1, b = 10, c = 26} \\ h\text{ = }(-10)/(2(1))\text{ = -5} \\ \\ k\text{ = f\lparen h\rparen = f\lparen-5\rparen} \\ f(-5)\text{ = \lparen-5\rparen}^2\text{ + 10\lparen-5\rparen + 26} \\ f(-5)\text{ = 25 - 50 + 26} \\ f(-5)\text{ = 1} \\ k\text{ = 1} \end{gathered}`

Hence, the vertex (h, k) = (-5, 1)

iv) y-intercept is the value of y when x is equal to zero

We can equate x = 0 nd solve for xy. Or we could use the graph to find y-intercept

The value of y when the line crosses the y-axis will be the intercept

f(x) = x² + 10x + 26

when x = 0

f(x) = (0)² + 10(0) + 26

f(x) = 26

Hence, the y-intercept is 26

v) The x-intercept is the value of x when y is zero. To solve this algebraically, we will equate f(x) to zero and solve for y. Using a graph, it will be the value of x when the line crosses the x-axis.

From the graph, the line doesn't touch the x axis. This means there are no intercept (no intercept where its vale is a real number)