Problem 81 Find the vertex of the graph of ... [FREE SOLUTION]

Chapter 8: Problem 81

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find the $y$ -intercept, approximate the $x$ -intercepts to one decimal place, and sketch the graph. $$ f(x)=3 x^{2}-6 x+7 $$

Short Answer

Expert verified

Vertex: $(1, 4)$, opens upward, $y$-intercept: $7$, no real $x$-intercepts.

Step by step solution

Identify the Coefficients

Given the quadratic function $f(x) = 3x^2 - 6x + 7$, identify the coefficients: $a = 3$, $b = -6$, and $c = 7$.

Find the Vertex

The vertex $(h, k)$ of a parabola in the form $ax^2 + bx + c$ can be found using the formula $h = -\frac{b}{2a}$. Substitute the values to get $h = -\frac{-6}{2 \times 3} = 1$. Then calculate $k$ by substituting $x = 1$ into the original function: $k = 3(1)^2 - 6(1) + 7 = 4$. Thus, the vertex is $(1, 4)$.

Determine the Direction the Parabola Opens

Since the coefficient $a = 3$ is positive, the parabola opens upward.

Find the Y-Intercept

The $y$-intercept occurs when $x = 0$. Substitute $x = 0$ into the quadratic function: $f(0) = 3(0)^2 - 6(0) + 7 = 7$. Thus, the $y$-intercept is $(0, 7)$.

Approximate the X-Intercepts

The $x$-intercepts can be found by solving the equation $3x^2 - 6x + 7 = 0$. Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Calculate the discriminant: $b^2 - 4ac = (-6)^2 - 4 \cdot 3 \cdot 7 = 36 - 84 = -48$. Since the discriminant is negative, there are no real $x$-intercepts.

Sketch the Graph

To sketch the graph, mark the vertex $(1, 4)$ and the $y$-intercept $(0, 7)$ on the coordinate plane. Note that since the parabola opens upward and there are no real $x$-intercepts, the graph will not cross the $x$-axis. Draw the parabola based on this information.

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Key Concepts

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

Vertex of a Parabola

The vertex of a parabola is a critical point that signifies both the peak and the bottom of the curve. It is found in quadratic functions written in the standard form $ ax^2 + bx + c $. To find the vertex, you use the vertex formula $ h = -\frac{b}{2a} $ for the $ x $-coordinate, $ h $, and then calculate the $ y $-coordinate, $ k $, by substituting $ h $ back into the original equation.

First, identify the coefficients: $ a, b, $ and $ c $.
Apply the formula $ h = -\frac{b}{2a} $ to find the $ x $-coordinate.
Substitute $ x = h $ back into the function to get the $ y $-coordinate $ k $.

In our example with the function $ f(x) = 3x^2 - 6x + 7 $, identifying $ a = 3 $, $ b = -6 $, gives a vertex of $ (1, 4) $. This point is where the parabola attains its minimum value, as the parabola opens upwards.

Y-intercept

The $ y $-intercept is the point where a graph intersects the $ y $-axis. In other words, it is the value of $ f(x) $ when $ x = 0 $. This is a simple substitution process.

Set $ x = 0 $ in the quadratic equation.
Calculate the resulting $ y $ value to find the $ y $-intercept.

For $ f(x) = 3x^2 - 6x + 7 $, substituting $ x = 0 $ yields $ f(0) = 7 $. So, the $ y $-intercept is at the point $ (0, 7) $. This intercept point serves as one of the key markers on the graph and helps to sketch the curve accurately.

Quadratic Formula

The quadratic formula is an essential tool for finding the roots or $ x $-intercepts of quadratic functions. It is typically used when straightforward factoring is challenging. The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's how you use it:

Calculate the discriminant $ b^2 - 4ac $.
If the discriminant is positive, there are two real solutions; if zero, one real solution; and if negative, complex (no real) solutions.

For the function $ f(x) = 3x^2 - 6x + 7 $, the discriminant is $ -48 $. Since it is negative, there are no real $ x $-intercepts. Thus, the parabola does not intersect the $ x $-axis in the graph.

Graph of a Quadratic Function

Graphing a quadratic function involves several steps yet reveals much about the nature of the function. Begin by identifying the vertex and intercepts. The function $ f(x) = 3x^2 - 6x + 7 $ provides a vertex at $ (1, 4) $ and a $ y $-intercept at $ (0, 7) $.
The sign of the coefficient $ a $ determines the parabola's direction:

If $ a > 0 $, the parabola opens upwards.
If $ a < 0 $, it opens downwards.

Since $ a = 3 $ is positive, this parabola opens upwards.
To sketch the curve:

Plot the vertex and $ y $-intercept points.
Use the opening direction and symmetry about the vertex to sketch the rest of the curve.

The absence of real $ x $-intercepts in this case means the parabola never crosses the $ x $-axis, and smoothly curves above it.

91影视

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find the \(y\) -intercept, approximate the \(x\) -intercepts to one decimal place, and sketch the graph. $$ f(x)=3 x^{2}-6 x+7 $$

Short Answer

Step by step solution

Identify the Coefficients

Find the Vertex

Determine the Direction the Parabola Opens

Find the Y-Intercept

Approximate the X-Intercepts

Sketch the Graph

Key Concepts

Vertex of a Parabola

Y-intercept

Quadratic Formula

Graph of a Quadratic Function

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