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Chapter 3: Problem 9

For each of the follow quadratics, find a) the vertex, b) the vertical intercept, and \(c\) ) the horizontal intercepts. $$ f(x)=2 x^{2}-10 x+4 $$

Short Answer

Expert verified
Vertex is (2.5, -8.5), vertical intercept is (0, 4), horizontal intercepts are \(\left(\frac{5 + \sqrt{17}}{2}, 0\right)\) and \(\left(\frac{5 - \sqrt{17}}{2}, 0\right)\).

Step by step solution

01

Finding the Vertex

The vertex of a quadratic function in the form \(ax^2 + bx + c\) can be found using the formula \(x = -\frac{b}{2a}\). For the quadratic \(f(x) = 2x^2 - 10x + 4\), we identify \(a = 2\) and \(b = -10\). Substituting these values gives:\[x = -\frac{-10}{2 \times 2} = \frac{10}{4} = 2.5\]To find the \(y\)-coordinate of the vertex, substitute \(x = 2.5\) back into the original function:\[f(2.5) = 2(2.5)^2 - 10(2.5) + 4 = 2(6.25) - 25 + 4 = 12.5 - 25 + 4 = -8.5\]Thus, the vertex is \((2.5, -8.5)\).
02

Finding the Vertical Intercept

The vertical intercept (or \(y\)-intercept) occurs when \(x = 0\). Substitute \(x = 0\) into the function \(f(x) = 2x^2 - 10x + 4\):\[f(0) = 2(0)^2 - 10(0) + 4 = 4\]Therefore, the vertical intercept is \((0, 4)\).
03

Finding the Horizontal Intercepts

To find the horizontal intercepts (or \(x\)-intercepts), set the function equal to zero and solve for \(x\):\[2x^2 - 10x + 4 = 0\]We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = -10\), and \(c = 4\).First, calculate the discriminant:\[b^2 - 4ac = (-10)^2 - 4(2)(4) = 100 - 32 = 68\]Now, solve for \(x\):\[x = \frac{10 \pm \sqrt{68}}{4} = \frac{10 \pm 2\sqrt{17}}{4} = \frac{5 \pm \sqrt{17}}{2}\]Thus, the horizontal intercepts are \(\left(\frac{5 + \sqrt{17}}{2}, 0\right)\) and \(\left(\frac{5 - \sqrt{17}}{2}, 0\right)\).

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Key Concepts

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

Vertex Form
The vertex of a quadratic function is a point on the graph that represents either the highest or the lowest point, depending on the orientation of the parabola. A quadratic function can often be rewritten in the vertex form:
  • Vertex form: \( y = a(x - h)^2 + k \)
  • Where \((h, k)\) is the vertex of the parabola
For the given quadratic function \( f(x) = 2x^2 - 10x + 4 \), we used the formula \( x = -\frac{b}{2a} \) to find the vertex:
  • \( a = 2 \), \( b = -10 \)
  • Substitute into the formula: \( x = \frac{10}{4} = 2.5 \)
This tells us that the x-coordinate of the vertex is 2.5. To find the corresponding y-coordinate, substitute \( x = 2.5 \) into the function:
  • \( f(2.5) = 2(2.5)^2 - 10(2.5) + 4 = -8.5 \)
Therefore, the vertex is located at the point \((2.5, -8.5)\). This point gives us valuable information about the vertex form of the parabola.
Vertical Intercept
The vertical intercept of a quadratic function is the point where the graph crosses the y-axis. Since the y-axis is where \( x = 0 \), you can find the vertical intercept by setting \( x = 0 \) in the function.
For the quadratic \( f(x) = 2x^2 - 10x + 4 \), we calculate:
  • \( f(0) = 2(0)^2 - 10(0) + 4 = 4 \)
This means the vertical intercept is \((0, 4)\).
The vertical intercept provides essential information about the constant term, \( c \), in the function, which directly tells us where the graph meets the y-axis.
It's a key point that helps in sketching the graph of the function.
Horizontal Intercepts
Horizontal intercepts are the points where the graph of the quadratic crosses or touches the x-axis. These points occur when the value of \( f(x) \) is zero (i.e., \( y = 0 \)).
To find them, you solve the equation \( 2x^2 - 10x + 4 = 0 \) for \( x \).We use the quadratic formula:
  • \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
  • For our function: \( a = 2 \), \( b = -10 \), \( c = 4 \)
  • Calculate the discriminant: \( b^2 - 4ac = 68 \)
  • Find \( x \): \( x = \frac{10 \pm \sqrt{68}}{4} = \frac{5 \pm \sqrt{17}}{2} \)
Thus, the horizontal intercepts are \( \left(\frac{5 + \sqrt{17}}{2}, 0\right) \) and \( \left(\frac{5 - \sqrt{17}}{2}, 0\right) \).
These intercepts are crucial for understanding where the parabola touches or crosses the x-axis, which helps in graphing and analyzing the function's behavior.
Quadratic Formula
The quadratic formula is a powerful tool used to find the solutions or roots of a quadratic equation. This formula solves equations of the form \( ax^2 + bx + c = 0 \):
  • Quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
  • "\(+\sqrt{...}\)" and "\(-\sqrt{...}\)" represent the possible solutions
The expression \( b^2 - 4ac \) is called the discriminant.
For \( f(x) = 2x^2 - 10x + 4 \):
  • Discriminant \( 68 \) indicates two real solutions
  • Solutions: \( x = \frac{5 \pm \sqrt{17}}{2} \)
The quadratic formula not only provides solutions but also gives insights into the nature of these solutions based on the discriminant:
  • Positive discriminant: Two distinct real solutions
  • Zero discriminant: One real solution
  • Negative discriminant: No real solutions
It's an essential method for finding intercepts and understanding the quadratic function's roots.

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