Problem 72 Determine the coordinates of the... [FREE SOLUTION]

Chapter 10: Problem 72

Determine the coordinates of the vertex of the graph of each function using the vertex formula. Then determine the $x-$ and $y$ -intercepts of the graph. Finally, plot several points and complete the graph. See Example $9 .$ $$ f(x)=-2 x^{2}+8 x-10 $$

Short Answer

Expert verified

The vertex is $(2, -2)$, $ y $-intercept is $(0, -10)$. No real $ x $-intercepts.

Step by step solution

Identify Parameters of the Quadratic Function

The given quadratic function is in the form $ f(x) = ax^2 + bx + c $, where $ a = -2 $, $ b = 8 $, and $ c = -10 $. Identify these coefficients for later use in the vertex formula.

Use Vertex Formula to Find Vertex Coordinates

The vertex $(h, k)$ of a quadratic function $ f(x) = ax^2 + bx + c $ can be found using the formula: $ h = -\frac{b}{2a} $ and $ k = f(h) $. Substitute the values: $ h = -\frac{8}{2(-2)} = 2 $. Calculate $ k $ by substituting $ h $ into the function: $ k = f(2) = -2(2)^2 + 8(2) - 10 = -8 + 16 - 10 = -2 $. So, the vertex is $(2, -2)$.

Find the X-Intercepts of the Function

To find the $ x $-intercepts, set $ f(x) = 0 $: $-2x^2 + 8x - 10 = 0$. Simplify by dividing the entire equation by $-2$: $x^2 - 4x + 5 = 0$. Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $ a = 1 $, $ b = -4 $, $ c = 5 $. Calculate the discriminant: $ (-4)^2 - 4(1)(5) = 16 - 20 = -4 $. Since the discriminant is negative, there are no real $ x $-intercepts.

Find the Y-Intercept of the Function

The $ y $-intercept occurs when $ x = 0 $. Substitute $ x = 0 $ into the function: $ f(0) = -2(0)^2 + 8(0) - 10 = -10 $. Thus, the $ y $-intercept is $ (0, -10) $.

Plot Points and Sketch the Graph

In addition to the vertex $(2, -2)$ and the $ y $-intercept $(0, -10)$, choose a few more points to better sketch the parabola. For example: $ f(1) = -2(1)^2 + 8(1) - 10 = -4 $ and $ f(3) = -2(3)^2 + 8(3) - 10 = -4 $. Plot these points: $ (1, -4) $ and $ (3, -4) $. Draw a symmetrical parabola opening downwards through these points.

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Key Concepts

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

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of a quadratic equation, which is any equation that can be rewritten in the form $ ax^2 + bx + c = 0 $. This formula gives us the roots or the $ x $-intercepts by solving for $ x $.

It is given by:

$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $

Here, $ a $, $ b $, and $ c $ are coefficients from the quadratic equation. The term $ b^2 - 4ac $ is known as the discriminant. The discriminant tells us the nature of the roots:

If it is positive, there are two distinct real roots.
If it is zero, there is exactly one real root (also called a repeated or double root).
If it is negative, the roots are complex and not real 鈥� this means the parabola does not cross the $ x $-axis.

In our example, for the function $ f(x) = -2x^2 + 8x - 10 $, we found that the discriminant was negative, indicating no real $ x $-intercepts.

X-Intercepts

X-intercepts are points where the graph of the function crosses the $ x $-axis. At these points, the value of $ y $ or $ f(x) $ is zero. To find the $ x $-intercepts, we set the quadratic function equal to zero and solve for $ x $.

In mathematical terms:

Set $ f(x) = 0 $

Using the example $ f(x) = -2x^2 + 8x - 10 $, setting this equal to zero would give us a quadratic equation that we solve using the quadratic formula.
Since the discriminant $ (b^2 - 4ac) $, calculated as $(-4)^2 - 4(1)(5)$, is negative, it indicates no real solutions. Therefore, this function has no $ x $-intercepts. This tells us that the parabola does not cut the $ x $-axis at any point.

Y-Intercepts

The $ y $-intercepts are found by evaluating the function when $ x = 0 $. This will tell us where the graph crosses the $ y $-axis.

To find the $ y $-intercept, substitute $ x = 0 $ in the function:

$ f(0) = -2(0)^2 + 8(0) - 10 = -10 $

The $ y $-intercept is therefore at the point $(0, -10)$.
This means that when the function's graph is plotted, it will cross the $ y $-axis at $-10$. The $ y $-intercept provides a starting point for sketching the graph and helps us understand how the parabola is positioned along the $ y $-axis.

Vertex Formula

Finding the vertex of a quadratic function is crucial as it represents the highest or lowest point of the parabola, depending on whether it opens upwards or downwards. This point is also a key part of sketching the graph.

The vertex $(h, k)$ can be found using the vertex formula:

$ h = -\frac{b}{2a} $
$ k = f(h) $

For $ f(x) = -2x^2 + 8x - 10 $:

First, calculate $ h = -\frac{8}{2(-2)} = 2 $.
Then, find $ k $ by substituting $ h $ back into the function: $ k = f(2) = -2(2)^2 + 8(2) - 10 = -2 $.

So, our vertex is $(2, -2)$.
The vertex gives us vital information about the graph. Since the quadratic's leading coefficient $ a $ is negative, the parabola opens downwards, making the vertex a maximum point on the graph.

91影视

Determine the coordinates of the vertex of the graph of each function using the vertex formula. Then determine the \(x-\) and \(y\) -intercepts of the graph. Finally, plot several points and complete the graph. See Example \(9 .\) $$ f(x)=-2 x^{2}+8 x-10 $$

Short Answer

Step by step solution

Identify Parameters of the Quadratic Function

Use Vertex Formula to Find Vertex Coordinates

Find the X-Intercepts of the Function

Find the Y-Intercept of the Function

Plot Points and Sketch the Graph

Key Concepts

Quadratic Formula

X-Intercepts

Y-Intercepts

Vertex Formula

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