Problem 45 Find the vertex and intercepts f... [FREE SOLUTION]

Chapter 8: Problem 45

Find the vertex and intercepts for each quadratic function. Sketch the graph, and state the domain and range. $$g(x)=x^{2}+2 x-8$$

Short Answer

Expert verified

Vertex: $ (-1, -9) $, x-intercepts: $ -4 $ and $ 2 $, y-intercept: $ (0, -8) $, domain: $ (-\infty, \infty) $, range: $ [-9, \infty) $.

Step by step solution

Identify the quadratic function

The given quadratic function is $ g(x) = x^2 + 2x - 8 $.

Find the vertex

The vertex form of a quadratic function is given by $ g(x) = a(x - h)^2 + k $, where $ h $ and $ k $ are the coordinates of the vertex.For the standard form $ ax^2 + bx + c $, the vertex is found using $ h = -\frac{b}{2a} $. Here, $ a = 1 $, $ b = 2 $, and $ c = -8 $. So, $ h = -\frac{2}{2(1)} = -1 $. To find $ k $, substitute $ h $ back into the function: $ g(-1) = (-1)^2 + 2(-1) - 8 = 1 - 2 - 8 = -9 $. Thus, the vertex is $ (-1, -9) $.

Find the x-intercepts

To find the x-intercepts, set $ g(x) = 0 $ and solve for $ x $: $ x^2 + 2x - 8 = 0 $. This can be factored as $ (x + 4)(x - 2) = 0 $. Thus, $ x + 4 = 0 $ or $ x - 2 = 0 $, giving $ x = -4 $ and $ x = 2 $ as the x-intercepts.

Find the y-intercept

To find the y-intercept, set $ x = 0 $ and solve for $ g(x) $: $ g(0) = 0^2 + 2(0) - 8 = -8 $. So, the y-intercept is $ (0, -8) $.

Determine the domain

The domain of any quadratic function is all real numbers. Thus, the domain is $ (-\infty, \infty) $.

Determine the range

Since the parabola opens upwards (a positive leading coefficient), the range starts from the $ k $-value of the vertex and goes to infinity. Therefore, the range is $ [-9, \infty) $.

Sketch the graph

Plot the vertex $ (-1, -9) $, the x-intercepts $ (-4, 0) $ and $ (2, 0) $, and the y-intercept $ (0, -8) $, then draw the parabola opening upwards.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

vertex of a parabola

In a quadratic function, the vertex is a key point. It is found using the formula for the standard form quadratic, which is $ax^2 + bx + c$. The vertex is given by the coordinates $h = -\frac{b}{2a}$ and $k = g(h)$. In our example $g(x) = x^2 + 2x - 8$, the values are $a = 1$, $b = 2$, and $c = -8$. Plugging in these values:

$h = -\frac{2}{2(1)} = -1$.

Then calculate $k$ by substituting $h$ into the function: $g(-1) = (-1)^2 + 2(-1) - 8 = -9$.

Therefore, the vertex is $(-1, -9)$. The vertex is where the parabola changes direction.

x-intercepts

X-intercepts are points where the parabola crosses the x-axis. To find them, set the quadratic equation to zero: $g(x) = 0$. For $g(x) = x^2 + 2x - 8$:

Firstly, solve $x^2 + 2x - 8 = 0$.
Factor the quadratic: $(x + 4)(x - 2) = 0$.
This gives solutions $x = -4$ and $x = 2$.

So, the x-intercepts are $(-4, 0)$ and $(2, 0)$. X-intercepts are useful as they show where the function value is zero.

y-intercept

The y-intercept is the point where the graph crosses the y-axis. To find the y-intercept, set $x$ to zero and solve for $g(x)$:

For $g(x) = x^2 + 2x - 8$, calculate $g(0)$:
$g(0) = 0^2 + 2(0) - 8 = -8$.

Thus, the y-intercept is $(0, -8)$. The y-intercept tells us the value of the function when $x = 0$.

domain and range

Understanding the domain and range of a quadratic function helps to know where the function is defined and how it behaves.

Domain: For any quadratic function, the domain is all real numbers. So, $ (-\infty, \, \infty) $.
Range: Depending on the direction the parabola opens (upwards or downwards), the range changes. Here, since $g(x) = x^2 + 2x - 8$ opens upwards (positive leading coefficient), the range starts from the vertex's $y$-value and goes to infinity. Thus, the range is $ [-9, \, \infty) $.

parabola graphing

Graphing a parabola puts everything together. To sketch $ g(x) = x^2 + 2x - 8 $, follow these steps:

Plot the vertex $(-1, -9)$.
Mark the x-intercepts $(-4, 0)$ and $(2, 0)$.
Plot the y-intercept $(0, -8)$.

After marking these points, draw a smooth curve through them opening upwards. The vertex is the lowest point. The graph shows visually how the parabola behaves.

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91影视

Find the vertex and intercepts for each quadratic function. Sketch the graph, and state the domain and range. $$g(x)=x^{2}+2 x-8$$

Short Answer

Step by step solution

Identify the quadratic function

Find the vertex

Find the x-intercepts

Find the y-intercept

Determine the domain

Determine the range

Sketch the graph

Key Concepts

vertex of a parabola

x-intercepts

y-intercept

domain and range

parabola graphing

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