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Chapter 3: Problem 8

For each of the follow quadratics, find a) the vertex, b) the vertical intercept, and \(c\) ) the horizontal intercepts. $$ z(p)=3 x^{2}+6 x-9 $$

Short Answer

Expert verified
Vertex: \((-1, -12)\); Vertical intercept: \((0, -9)\); Horizontal intercepts: \((-3, 0)\), \((1, 0)\).

Step by step solution

01

Rewrite the Quadratic in Standard Form

The given quadratic is \(z(p)=3x^{2}+6x-9\). It is already in the standard form \(ax^2 + bx + c\), where \(a = 3\), \(b = 6\), and \(c = -9\).
02

Find the Vertex

To find the vertex of a quadratic \(ax^2 + bx + c\), use the formula for the vertex \((h, k)\), where \(h = -\frac{b}{2a}\). Substitute \(a = 3\) and \(b = 6\):\[h = -\frac{6}{2(3)} = -1\]Then find \(k\) by substituting \(h\) back into the equation:\[k = 3(-1)^2 + 6(-1) - 9 = 3 - 6 - 9 = -12\]Thus, the vertex is \((-1, -12)\).
03

Find the Vertical Intercept

The vertical intercept occurs where \(x = 0\). Substitute \(x = 0\) into the equation:\[z(p) = 3(0)^2 + 6(0) - 9 = -9\]So, the vertical intercept is \((0, -9)\).
04

Find the Horizontal Intercepts

The horizontal intercepts occur where \(z(x) = 0\). Solve the equation:\[3x^2 + 6x - 9 = 0\]To simplify, divide the entire equation by 3:\[x^2 + 2x - 3 = 0\]Factor the quadratic:\[(x + 3)(x - 1) = 0\]Therefore, the solutions are:\[x + 3 = 0 \Rightarrow x = -3\]\[x - 1 = 0 \Rightarrow x = 1\]The horizontal intercepts are at \((-3, 0)\) and \((1, 0)\).

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Key Concepts

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

Vertex of a Quadratic
The vertex of a quadratic function is a key point that provides important information about the graph of the quadratic. In a quadratic equation of the form \(ax^2 + bx + c\), the vertex is found using the formula \((h, k)\), where \(h = -\frac{b}{2a}\). This formula gives us the x-coordinate of the vertex. Once we have \(h\), we can find the corresponding y-coordinate \(k\) by substituting \(h\) back into the quadratic equation.

Let's look at the equation given: \(z(p)=3x^{2}+6x-9\).
  • Identify \(a = 3\), \(b = 6\), \(c = -9\).
  • Calculate \(h = -\frac{6}{2(3)} = -1\).
  • Substitute \(h = -1\) into the equation to find \(k\): \(3(-1)^2 + 6(-1) - 9 = -12\).
Thus, the vertex of the quadratic is \((-1, -12)\). The vertex is the lowest point on the graph if it opens upwards, or the highest if it opens downwards.
Vertical Intercept
The vertical intercept of a quadratic equation is where the graph of the function crosses the y-axis. This occurs when the x-value is zero. Mathematically, it represents the point \((0, c)\) on the graph, where \(c\) is the constant term in the standard quadratic form \(ax^2 + bx + c\).
For the quadratic \(z(p) = 3x^{2}+6x-9\), finding the vertical intercept is straightforward:
  • Set \(x = 0\).
  • Calculate: \(z(p) = 3(0)^2 + 6(0) - 9 = -9\).
This means the vertical intercept of the quadratic function is at the point \((0, -9)\).
This point provides the y-intercept, showing where the parabola crosses the y-axis when plotted.
Horizontal Intercepts
Horizontal intercepts, also known as x-intercepts, are the points where a quadratic graph crosses the x-axis. To find these, we solve the equation \(z(x) = 0\). For the quadratic \(z(p) = 3x^2 + 6x - 9\), the steps are straightforward:
  • Set the equation to zero: \(3x^2 + 6x - 9 = 0\).
  • Simplify by dividing the entire equation by 3: \(x^2 + 2x - 3 = 0\).
  • Factor the quadratic: \((x + 3)(x - 1) = 0\).
Solving for \(x\), we find:
  • \(x + 3 = 0 \Rightarrow x = -3\)
  • \(x - 1 = 0 \Rightarrow x = 1\)
Thus, the horizontal intercepts are at \((-3, 0)\) and \((1, 0)\). These points indicate where the quadratic intersects the x-axis, as shown by its roots or solutions.
Factoring Quadratics
Factoring quadratics is an essential technique for solving quadratic equations and finding the x-intercepts. Factoring involves expressing a quadratic like \(ax^2 + bx + c\) as a product of two binomials.
In our example, \(z(p) = 3x^2 + 6x - 9\), we focus first on simplifying:
  • Divide each term by the greatest common factor, which is 3: \(x^2 + 2x - 3 = 0\).
Then, apply factoring:
  • Look for two numbers that multiply to \(-3\) (the constant term) and add to \(2\) (the linear coefficient). These numbers are \(3\) and \(-1\).
  • Write the quadratic as \((x + 3)(x - 1) = 0\).
Factoring reveals the solutions or roots \(x = -3\) and \(x = 1\), which are also the horizontal intercepts. Through factoring, quadratics can be solved more intuitively by breaking them into simpler parts.

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