Question

Consider the function f(x)=x^2+2x-8

1) what are the x intercepts of the graph of the function?
2) what is the y intercepts of the graph of the function?
3) what is the equation of the axis of symmetry?
4) what is the vertex of the function?
5) graph the function

Answer 1

1) The x-intercepts of the function
`\( f(x) = x^2 + 2x - 8 \)`are
`\( x = -4 \)`and
`\( x = 2 \).`

2) The y-intercept of the function
`\( f(x) = x^2 + 2x - 8 \) is \( y = -8 \).`

3) The equation of the axis of symmetry is
`\( x = -1 \).`

4) The vertex of the function
`\( f(x) = x^2 + 2x - 8 \) is \( (-1, -9) \).`

5) The graph of the function is a parabola opening upwards with the vertex at
`\( (-1, -9) \).`

Step-by-step explanation:

1) To find the x-intercepts, set
`\( f(x) = 0 \)` and solve for x. The quadratic equation
`\( x^2 + 2x - 8 = 0 \)` factors into
`\( (x - 2)(x + 4) = 0 \),` yielding x-intercepts of
`\( x = -4 \)` and
`\( x = 2 \).`

2) To find the y-intercept, set \( x = 0 \) in the function. \( f(0) = 0^2 + 2(0) - 8 = -8 \), so the y-intercept is
`\( y = -8 \).`

3) The axis of symmetry for a quadratic function in the form
`\( f(x) = ax^2 + bx + c \)` is given by
`\( x = (-b)/(2a) \).` For
`\( f(x) = x^2 + 2x - 8 \)`, the axis of symmetry is
`\( x = -1 \).`

4) The vertex of a quadratic function in the form
`\( f(x) = ax^2 + bx + c \) is located at \( \left(-(b)/(2a), f\left(-(b)/(2a)\right)\right) \).`Substituting
`\( x = -1 \)` into the function, we find that the vertex is
`\( (-1, -9) \).`

5) The graph of the function is a parabola that opens upwards, consistent with the positive coefficient of the
`\( x^2 \)` term. The vertex at (-1, -9) is the lowest point on the graph, and the parabola extends upward indefinitely from there.

Answer 2

The answer to your question is below

Step-by-step explanation:

Data

function f(x) = x² + 2x - 8

-See the graph below

1) x-intercepts

We observe in the graph that there are no x-intercepts.

2) y-intercepts

We observe in the graph that the y-intercept is (0, 8). Point A

3) axis of symmetry

The axis of symmetry is the line that divides the parabola into two equal parts. This line is x = -1 (blue line)

4) The vertex is the lowest point of the parabola, this point is (-1, 7)

Point B in the graph.