Intercepts
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This document explains how to find the x-intercept and y-intercept of a graph from its equation. The x-intercept is where the graph crosses the x-axis and is of the form (x,0), while the y-intercept is where the graph crosses the y-axis and is of the form (0,y). To find the x-intercept, set y=0 and solve for x; to find the y-intercept, set x=0 and solve for y. Examples are provided finding the intercepts of equations by setting variables equal to 0 and solving.
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ax2 + bx + c = 0
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2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
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1) This document discusses how to solve quadratic equations by graphing, including identifying the terms of a quadratic equation, finding the solutions by graphing, and graphing quadratic functions.
2) The key steps for graphing a quadratic function are to find the axis of symmetry using the standard form equation, find the vertex point, and find two other points to reflect across the axis of symmetry to complete the parabolic graph.
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1. Quadratic Equation
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ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
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7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
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Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
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Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
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Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
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Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
By Completing the Square
1. x² + 6x + 5 = 0
2. x² - 8x + 3 = 0
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2. It provides examples of writing equations in each form given specific information like two points, an x-intercept and y-intercept, a point and slope, or just a slope and y-intercept.
3. Guided practice problems are included for writing equations parallel or perpendicular to a given line.
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1) Write both equations in standard form Ax + By = C
2) Determine which variable to eliminate using addition or subtraction
3) Solve the resulting equation for one variable
4) Substitute back into the original equation to solve for the other variable
5) Check that the solution satisfies both original equations
It provides examples showing how to set up and solve systems of equations using elimination, including word problems about supplementary angles and finding two numbers based on their sum and difference.
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The document discusses trigonometric functions on the unit circle. It defines trig ratios for angles in each of the four quadrants using right triangles formed with the point (x,y) and the origin. Key identities presented are:
1) tanθ = sinθ/cosθ
2) sin2θ + cos2θ = 1
The signs of the trig functions depend on the quadrant, with trig ratios being positive in Quadrant I and changing appropriately in other quadrants based on the signs of x and y.
Factoring by grouping
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Can be used by a math teacher in her discussion about factoring
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1) Identify common factors in the numerator and denominator and divide them out.
2) Factor the numerator and denominator by identifying common factors in each expression.
3) Divide the factored numerator and denominator, simplifying the expression.
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The document provides instruction on linear functions and how to find the x-intercept and y-intercept of a linear equation. It discusses writing linear equations in function notation as f(x), defines the x-intercept as where the graph crosses the x-axis (y = 0) and the y-intercept as where it crosses the y-axis (x = 0). It gives the formula for linear equations in standard form and shows how to set x = 0 and y = 0 to find the intercepts. Examples are worked out, showing how to graph the line based on the intercepts. Classwork assignments are noted at the end.
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Intercepts
- 1. Intercepts HOW TO FIND THEM USING EQUATIONS AND GRAPHS
- 2. An intercept is where a graph crosses an axis. An x-intercept is where a graph crosses the x-axis. A y-intercept is where a graph crosses the y-axis.
- 3. Properties of Intercepts An x-intercept is always of the form (x, 0), because the y-coordinate must be 0 for the point to sit on the x-axis. A y-intercept is always of the form (0, y), because the x-coordinate must be 0 for the point to sit on the y- axis.
- 4. To find intercepts from an equation: To find the x-intercept of an equation, set y = 0 and solve for x. For example, let’s find the x-intercept of the graph of 3x – 2y = 12. -Plug in 0 for y: 3x – 2(0) = 12 -Solve for x: 3x – 0 = 12 3x = 12 x = 4 The x-intercept is located at (4, 0). To find the y-intercept of an equation, set x = 0 and solve for y. For example, let’s find the y-intercept of the graph of 3x – 2y = 12. -Plug in 0 for x: 3(0) – 2y = 12 -Solve for x: 0 – 2y = 12 -2y = 12 y = -6 The y-intercept is located at (0, -6).
- 5. You should be able to verify your intercepts by looking at the graph of an equation. Let’s solve for y in the previous example so we can graph the equation using our graphing calculator: 3𝑥 − 2𝑦 = 12 −2𝑦 = −3𝑥 + 12 𝑦 = 3 2 𝑥 − 6 (0, -6) (4, 0)
- 6. Find the intercepts of 4y = 2x – 6. Let’s find the x-intercept first by plugging in 0 for y: 4(0) = 2x – 6 0 = 2x – 6 6 = 2x 3 = x The x-intercept is (3, 0). Let’s find the y-intercept next by plugging in 0 for x: 4y = 2(0) – 6 4y = 0 – 6 4y = -6 y = -1.5 The y-intercept is (0, -1.5)